# A fully diagonalized spectral method using generalized Laguerre functions on
the half line

Fu-jun Liu1 ,  Zhong-qing Wang2 and  Hui-yuan Li3

###### Abstract.

A fully diagonalized spectral method using generalized Laguerre functions is
proposed and analyzed for solving elliptic equations on the half line. We
first define the generalized Laguerre functions which are complete and
mutually orthogonal with respect to an equivalent Sobolev inner product. Then
the Fourier-like Sobolev orthogonal basis functions are constructed for the
diagonalized Laguerre spectral method of elliptic equations. Besides, a
unified orthogonal Laguerre projection is established for various elliptic
equations. On the basis of this orthogonal Laguerre projection, we obtain
optimal error estimates of the fully diagonalized Laguerre spectral method for
both Dirichlet and Robin boundary value problems. Finally, numerical
experiments, which are in agreement with the theoretical analysis, demonstrate
the effectiveness and the spectral accuracy of our diagonalized method.

###### Key words and phrases:

Spectral method, Sobolev orthogonal Laguerre functions, elliptic boundary
value problems, error estimates.

###### 2000 Mathematics Subject Classification:

76M22, 33C45, 35J25, 65L70

1Department of Mathematics, Shanghai Normal University, Shanghai, 200234,
China; School of Science, Henan Institute of Engineering, Zhengzhou, 451191,
China. Email<EMAIL_ADDRESS>
2School of Science, University of Shanghai for Science and Technology,
Shanghai, 200093, China. Email<EMAIL_ADDRESS>
3State Key Laboratory of Computer Science/Laboratory of Parallel Computing,
Institute of Software, Chinese Academy of Sciences, Beijing 100190, China.
Email<EMAIL_ADDRESS>
The first author was supported by Science and Technology Research Program of
Education Department of Henan Province (No. 13A110005); The second author was
supported in part by National Natural Science Foundation of China (No.
11571238) and the Research Fund for Doctoral Program of Higher Education of
China (No. 20133127110006); The third author was supported by National Natural
Science Foundation of China (Nos. 91130014, 11471312 and 91430216).

##  1\. Introduction

Spectral methods for solving partial differential equations on unbounded
domains have gained a rapid development during the last few decades. An
abundance of literature on this research topic has emerged, and their
underlying approximation approaches can be essentially classified into three
catalogues [4, 27]:

  * (i)

truncate an unbounded domain to a bounded one and solve the problem on the
bounded domain subject to artificial or transparent boundary conditions [22,
26];

  * (ii)

map the original problem on an unbounded domain to one on a bounded domain and
use classic spectral methods to solve the new problem [9]; or equivalently,
approximate the original problem by some non-classical functions mapped from
the classic orthogonal polynomials/functions on a bounded domain [2, 3, 7, 11,
12, 27, 31, 34];

  * (iii)

directly approximate the original problem by genuine orthogonal functions such
as Laguerre polynomials or functions on the unbounded domain [6, 10, 13, 14,
15, 16, 17, 18, 19, 20, 24, 30, 32, 33, 35].

The third approach is of particular interest to researchers, and has won an
increasing popularity in a broad class of applications, owing to its essential
advantages over other two approaches. These direct approximation schemes
constitute an initial step towards the efficient spectral methods, which admit
fast and stable algorithms for their efficient implementations.

As we know, the Fourier spectral method makes use of the eigenfunctions of the
Laplace operator which are orthogonal to each other with respect to the
Sobolev inner product involving derivatives, thus the corresponding algebraic
system is diagonal [4, 5, 25]. This fact together with the availability of the
fast Fourier transform (FFT) makes the Fourier spectral method be an ideal
approximation approach for differential equations with periodic boundary
conditions. Although the utilization of the genuine orthogonal
polynomials/functions in this direct approach usually leads to a highly sparse
(e.g., tri-diagonal, penta-diagonal) and well-conditioned algebraic system,
however, in many cases, people still want to get a set of Fourier-like basis
functions for a fully diagonalized algebraic system [28].

The main purpose of this paper is to construct the Fourier-like Sobolev
orthogonal basis functions [8, 21] for elliptic boundary value problems on the
half line $\Lambda=(0,\infty)$. For this purpose, we shall first extend the
definition of Laguerre polynomials
$\big{\\{}\mathcal{L}^{\alpha}_{k}(x)\big{\\}}_{k\geq 0}$ and Laguerre
functions
$\big{\\{}l^{\alpha,\beta}_{k}(x)=e^{-\frac{\beta}{2}x}\mathcal{L}^{\alpha}_{k}(\beta
x)\big{\\}}_{k\geq 0}$ for $\beta>0$ to allow $\alpha$ being any real number.
The resulting generalized Laguerre functions are proven to be the
eigenfunctions of certain high order Sturm-Liouville differential operators
(see Lemma 2.6 of this paper). Moreover, they are complete and mutually
orthogonal in $H^{r}_{x^{r+\alpha}}(\Lambda)$ for any nonnegative integer
$r>-\alpha-1$ with respect to an equivalent Sobolev inner product (see (2.23)
of this paper).

Since the problem is dependent on the inner product originated from the
coercive bilinear form of the elliptic equation, it does not necessarily
coincide with the equivalent Sobolev inner product, further efforts should be
paid to obtain the Fourier-like basis functions for a fully diagonalized
spectral approximation, in spite of the Sobolev orthogonality of
$\big{\\{}l^{\alpha,\beta}_{k}(x)\big{\\}}_{k\geq 0}$. Starting with
$\big{\\{}l^{-1,\beta}_{k}(x)\big{\\}}_{k\geq 0}$, stable and efficient
algorithms are then proposed to construct the Fourier-like basis functions for
the non-homogeneous Dirichlet and Robin boundary value problems of the second
order elliptic equations. In the sequel, both the exact solution and the
approximate solution can be represented as infinite and truncated Fourier
series in $\big{\\{}l^{-1,\beta}_{k}(x)\big{\\}}$, respectively. Although the
fully diagonalized spectral methods are studied for second order equations,
they can be readily generalized to solve $2r$-th order equations by starting
with $\big{\\{}l^{-r,\beta}_{k}(x)\big{\\}}$.

An ideal spectral approximation to differential equations may guarantee an
optimal error estimate in its convergence analysis. To match this requirement,
various orthogonal projections involving different orders of derivatives and
boundary conditions have been designed and studied case by case, which
frequently make the numerical analysis in spectral method a tedious task.
Moreover, the traditional routine to measure the approximation error is first
to establish the norm defined by a second-order self-adjoint differential
operator, and then estimate the upper bound of the approximation error with
the induced norms. However, this practical approach usually fails to
characterize the function space in which the orthogonal projection has an
optimal error estimate.

To conquer these difficulties, we need a unified definition of the orthogonal
spectral projections with a systematic numerical analysis. Fortunately, the
Sobolev orthogonality of the generalized Laguerre functions
$\big{\\{}l^{\alpha,\beta}_{k}(x)\big{\\}}$ with a negative integer
$\alpha=-n$ enables us to define the unified orthogonal projection
$\pi^{-n,\beta}_{N}$ from $H^{n}_{x^{r-n}}(\Lambda)$ to the finite
approximation space for all nonnegative integer $r\geq n$, ignoring the
specific value of $r$. More importantly, such an orthogonal projection
$\pi^{-n,\beta}_{N}$ interpolates the endpoint function values up to the
$(n-1)$-th derivative, i.e,
$\partial_{x}^{\ell}\pi^{-n,\beta}_{N}u(0)=\partial_{x}^{\ell}u(0)$ for any
$0\leq\ell\leq n-1$ and $N\geq n$. This endpoint interpolation property
ensures $u-\pi^{-n,\beta}_{N}u\in H_{0}^{n}(\Lambda)$, thus makes
$\pi^{-n,\beta}_{N}$ applicable to both the Dirichlet and Robin boundary value
problems, and available to multi-domain spectral methods. Besides, owing to
the clarity of the orthogonality structure of the generalised Laguerre
functions, one can not only derive an optimal order of the convergence for the
approximated function, but also get a generic characterization of the function
space where the orthogonal projection has an optimal error estimate.

Therefore, the second purpose of this paper is to establish such a unified
orthogonal Laguerre projection, and apply it to the convergence analysis on
the fully diagonalized Laguerre spectral method for both the Dirichlet and
Robin boundary value problems of second order elliptic equations.

The remainder of the paper is organized as follows. In Section 2, we first
make conventions on the frequently used notations, and then introduce
generalized Laguerre polynomials and functions with arbitrary index $\alpha$.
The fully diagonalized Laguerre spectral methods and the implementation of
algorithms are proposed in Section 3 for the Dirichlet and Robin boundary
value problems of second order elliptic equations. Section 4 is then devoted
to the convergence analysis of the unified orthogonal projection together with
our Laguerre spectral methods. Finally, numerical results are presented in
Section 5 to demonstrate the effectiveness and accuracy of the proposed
diagonalized Laguerre spectral methods, which are in agreement with our
theoretical predictions.

##  2\. Generalized Laguerre polynomials and functions

###  2.1. Notations and preliminaries

Let $\Lambda=(0,\infty)$ and ${\varpi}(x)$ be a weight function which is not
necessary in $L^{1}(\Lambda)$. We define

$L^{2}_{\varpi}({\Lambda})=\\{v~{}|~{}v\hbox{ is measurable on $\Lambda$
and~{}}\|v\|_{\varpi}<\infty\\},$

with the following inner product and norm,

$(u,v)_{\varpi}=\displaystyle\int_{\Lambda}u(x)v(x){\varpi}(x)dx,\quad\|v\|_{\varpi}=(v,v)^{\frac{1}{2}}_{\varpi},\quad\forall
u,v\in L_{\varpi}^{2}(\Lambda).$

For simplicity, we denote $\frac{d^{k}v}{dx^{k}}=\partial^{k}_{x}v$ and
$\frac{dv}{dx}=v^{\prime}.$ For any integer $m\geq 0$, we define

$H_{\varpi}^{m}(\Lambda)=\\{v~{}|~{}\partial_{x}^{k}v\in
L^{2}_{\varpi}(\Lambda),~{}0\leq k\leq m\\},$

with the following semi-norm and norm,

$|v|_{m,{\varpi}}=\|\partial_{x}^{m}v\|_{\varpi},\qquad\|v\|_{m,{\varpi}}=\Big{(}\displaystyle\sum_{k=0}^{m}|v|_{k,{\varpi}}^{2}\Big{)}^{\frac{1}{2}}.$

For any real $r>0,$ we define the space $H^{r}_{\varpi}(\Lambda)$ and its norm
$\|v\|_{r,{\varpi}}$ by function space interpolation as in [1]. In cases where
no confusion arises, ${\varpi}$ may be dropped from the notations whenever
${\varpi}(x)\equiv 1.$ Specifically, we shall use the weight functions
$w=w(x)=x$ and $w^{\alpha}=w^{\alpha}(x)=x^{\alpha}$ in the subsequent
sections.

We denote by $\mathbb{R}$ the collection of real numbers, by $\mathbb{N}_{0}$
and $\mathbb{Z}^{-}$ the collections of nonnegative and negative integers,
respectively. Further, we let $\mathbb{P}_{k}$ be the space of polynomials of
degree $\leq k$.

Let $\aleph:=\mathbb{Z}^{-}\cup(-1,+\infty)$. We also define the
characteristic functions $\chi_{n}$ for $n\in\mathbb{N}_{0}$,

$\displaystyle\chi_{n}(\alpha)=\begin{cases}-\alpha,&\alpha+n\in\mathbb{Z}^{-},\\\
0,&\alpha+n\in(-1,+\infty).\end{cases}$

For short we write $\chi(\alpha)=\chi_{0}(\alpha)$.

###  2.2. Generalized Laguerre polynomials

It is well known that, for $\alpha>-1,$ the classical Laguerre polynomials
${\mathcal{L}}_{k}^{{\alpha}}(x),\ k=0,1,\dots,$ admit an explicit
representation (see [29]):

${\mathcal{L}}_{k}^{{\alpha}}(x)=\displaystyle\sum_{\nu=0}^{k}\frac{(\alpha+\nu+1)_{k-\nu}}{(k-\nu)!\nu!}(-x)^{\nu},\qquad
x\in\Lambda,\quad k\geq 0,$ (2.1)

where we use the Pochhammer symbol $(a)_{n}=a(a+1)\dots(a+n-1)$ for any
$a\in\mathbb{R}$ and $n\in\mathbb{N}_{0}$.

The classical Laguerre polynomials can be extended to cases with any
$\alpha\in\mathbb{R}$ and the same representation as (2.1), which are referred
to as the generalized Laguerre polynomials (cf. [23]). Obviously, the
generalized Laguerre polynomials ${\mathcal{L}}_{k}^{{\alpha}}(x),\
k=0,1,\dots,$ constitute a complete basis for the linear space of real
polynomials as well, since $\deg{\mathcal{L}}^{\alpha}_{k}=k$ for all $k\geq
0$.

The generalized Laguerre polynomials fulfill the following recurrence
relations.

######  Lemma 2.1.

For any $\alpha\in\mathbb{R}$, it holds

$\left\\{\begin{array}[]{ll}{\mathcal{L}}^{\alpha}_{0}(x)=1,\qquad\qquad{\mathcal{L}}^{\alpha}_{1}(x)=-x+\alpha+1,\\\\[6.0pt]
(k+1){\mathcal{L}}^{\alpha}_{k+1}(x)=(2k+\alpha+1-x){\mathcal{L}}^{\alpha}_{k}(x)-(k+\alpha){\mathcal{L}}^{\alpha}_{k-1}(x),\qquad
k\geq 1.\end{array}\right.$ (2.2)

###### Proof.

The recurrence relation (2.2) for $\alpha\in\mathbb{R}$ can be derived from
those of the classic Laguerre polynomials for $\alpha>-1$ by the continuation
method. Here, we also give a concrete proof by the representation (2.1). Using
the expression (2.1), we obtain that for integer $k\geq 1,$

$\begin{array}[]{ll}(2k+\alpha+1){\mathcal{L}}^{\alpha}_{k}(x)-(k+\alpha){\mathcal{L}}^{\alpha}_{k-1}(x)-(k+1){\mathcal{L}}^{\alpha}_{k+1}(x)\\\\[6.0pt]
=(2k+\alpha+1)\displaystyle\sum_{\nu=0}^{k}\frac{(-1)^{\nu}\Gamma(k+\alpha+1)}{\nu!(k-\nu)!\Gamma(\alpha+\nu+1)}x^{\nu}-(k+\alpha)\displaystyle\sum_{\nu=0}^{k-1}\frac{(-1)^{\nu}\Gamma(k+\alpha)}{\nu!(k-\nu-1)!\Gamma(\alpha+\nu+1)}x^{\nu}\\\
\quad-(k+1)\displaystyle\sum_{\nu=0}^{k+1}\frac{(-1)^{\nu}\Gamma(k+\alpha+2)}{\nu!(k-\nu+1)!\Gamma(\alpha+\nu+1)}x^{\nu}.\end{array}$

Then a direct computation shows that

$\displaystyle(2k+\alpha$
$\displaystyle+1){\mathcal{L}}^{\alpha}_{k}(x)-(k+\alpha){\mathcal{L}}^{\alpha}_{k-1}(x)-(k+1){\mathcal{L}}^{\alpha}_{k+1}(x)$
$\displaystyle=$
$\displaystyle\,\displaystyle\sum_{\nu=1}^{k+1}\dfrac{(-1)^{\nu-1}\Gamma(k+\alpha+1)}{(\nu-1)!(k-\nu+1)!\Gamma(\alpha+\nu)}x^{\nu}=x{\mathcal{L}}^{\alpha}_{k}(x).$

The desired result is now derived. ∎

######  Lemma 2.2.

For any $\alpha\in\mathbb{R}$ and $k\geq 0$, it holds

$\displaystyle{\mathcal{L}}^{\alpha}_{k}(x)={\mathcal{L}}^{\alpha+1}_{k}(x)-{\mathcal{L}}^{\alpha+1}_{k-1}(x),$
(2.3)
$\displaystyle\partial_{x}{\mathcal{L}}^{\alpha}_{k}(x)=-{\mathcal{L}}^{\alpha+1}_{k-1}(x),$
(2.4) $\displaystyle
x\partial_{x}{\mathcal{L}}^{\alpha}_{k}(x)=k{\mathcal{L}}^{\alpha}_{k}(x)-(k+\alpha){\mathcal{L}}^{\alpha}_{k-1}(x),$
(2.5)
$\displaystyle{\mathcal{L}}^{\alpha}_{k}(x)=\partial_{x}{\mathcal{L}}^{\alpha}_{k}(x)-\partial_{x}{\mathcal{L}}^{\alpha}_{k+1}(x),$
(2.6)

where ${\mathcal{L}}^{\alpha}_{k}(x)\equiv 0$ for any $k\in\mathbb{Z}^{-}.$

###### Proof.

The recurrence relations (2.3)-(2.5) can be obtained readily by using similar
arguments as in Lemma 2.1. Moreover, by (2.3) and (2.4), it is easy to derive
(2.6). ∎

######  Lemma 2.3.

For any $\alpha\in\mathbb{R}$, the generalized Laguerre polynomials
${\mathcal{L}}^{\alpha}_{k}(x)$ satisfy the Sturm-Liouville equation

$x^{-\alpha}e^{x}\partial_{x}(x^{\alpha+1}e^{-x}\partial_{x}\mathcal{L}_{k}^{\alpha}(x))+\lambda_{k}\mathcal{L}_{k}^{\alpha}(x)=0,\quad
k\geq 0,$ (2.7)

or equivalently,

$x\partial_{x}^{2}\mathcal{L}_{k}^{\alpha}(x)+(\alpha+1-x)\partial_{x}\mathcal{L}_{k}^{\alpha}(x)+\lambda_{k}\mathcal{L}_{k}^{\alpha}(x)=0,\quad
k\geq 0,$ (2.8)

with the corresponding eigenvalue $\lambda_{k}=k$.

###### Proof.

Lemma 2.3 can be proved by the continuation method from the Sturm-Liouville
equation of the classic Laguerre polynomials for $\alpha>-1$. Also one can
give a proof by using the representation (2.1). We omit the details here. ∎

We are interested in those generalized Laguerre polynomials with an integer
index $\alpha\in\mathbb{Z}^{-}$.

######  Lemma 2.4.

For any $\alpha\in\mathbb{Z}^{-}$, we have

$\mathcal{L}_{k}^{\alpha}(x)=(-x)^{-\alpha}\frac{(k+\alpha)!}{k!}\mathcal{L}_{k+\alpha}^{-\alpha}(x),\qquad
k\geq\chi(\alpha).$ (2.9)

And for any $\alpha\in\aleph$, the following orthogonality relation holds:

$\displaystyle\int_{\Lambda}{\mathcal{L}}^{\alpha}_{k}(x){\mathcal{L}}^{\alpha}_{m}(x)x^{\alpha}e^{-x}dx=\gamma_{k}^{\alpha}\delta_{k,m},\quad\gamma_{k}^{\alpha}=\frac{\Gamma(k+\alpha+1)}{k!},\qquad
k,m\geq\chi(\alpha),$ (2.10)

where $\delta_{k,m}$ is the Kronecker symbol.

###### Proof.

The identity (2.9) comes directly from [29]. The orthogonality relation (2.10)
is known for classic Laguerre polynomials with $\alpha\in(-1,+\infty)$; while
for $\alpha\in\mathbb{Z}^{-}$, (2.10) can be obtained immediately from (2.9)
together with (2.10) for $\alpha\in(-1,+\infty)$. ∎

We now conclude this subsection with some generalized Laguerre polynomials
${\mathcal{L}}^{\alpha}_{k}(x)$ for $\alpha\in\mathbb{Z}^{-}$.

$\displaystyle\begin{array}[]{ccccccc}&k=0&k=1&k=2&k=3&\dots&k\geq\chi(\alpha)\\\
\hline\cr\alpha=-1&1&-x&\tfrac{1}{2}x(x-2)&-\tfrac{1}{6}x(x^{2}-6x+6)&\dots&-\frac{1}{k}x{\mathcal{L}}^{1}_{k-1}(x)\\\
\hline\cr\alpha=-2&1&-x-1&\tfrac{1}{2}x^{2}&-\tfrac{1}{6}x^{2}(x-3)&\dots&\frac{1}{k(k-1)}x^{2}{\mathcal{L}}^{2}_{k-2}(x)\\\
\hline\cr\alpha=-3&1&-x-2&\tfrac{1}{2}x^{2}+x+1&-\tfrac{1}{6}x^{3}&\dots&-\frac{1}{k(k-1)(k-2)}x^{3}{\mathcal{L}}^{3}_{k-3}(x)\\\
\hline\cr\dots&\dots&\dots&\dots&\dots&\dots&\dots\end{array}$

###  2.3. Generalized Laguerre functions

In this subsection, we shall introduce the generalized Laguerre functions with
arbitrary parameters $\alpha\in\mathbb{R}$ and $\beta>0$ and present some
properties.

The generalized Laguerre functions $l_{k}^{\alpha,\beta}(x),$ $k\geq 0$ are
defined by

$l_{k}^{\alpha,\beta}(x)=e^{-\frac{1}{2}{\beta}x}\mathcal{L}_{k}^{\alpha}(\beta
x),\qquad\forall\alpha\in\mathbb{R},\quad\beta>0,$ (2.11)

and the multiplication of $e^{-\frac{1}{2}\beta x}$ and the leading term of
$\mathcal{L}_{k}^{\alpha}(\beta x)$ is simply referred to as the leading term
of $l^{\alpha,\beta}_{k}(x)$.

According to (2.9), for any $\alpha\in\mathbb{Z}^{-}$, we have

$l_{k}^{\alpha,\beta}(x)=(-\beta
x)^{-\alpha}\frac{(k+\alpha)!}{k!}l_{k+\alpha}^{-\alpha,\beta}(x),\quad
k\geq\chi(\alpha),$ (2.12)

which means that $x=0$ is a zero of $l_{k}^{\alpha,\beta}(x)$ with the
multiplicity $-\alpha$, i.e.,

$\displaystyle\partial_{x}^{\nu}l_{k}^{\alpha,\beta}(0)=0,\quad
k\geq\chi(\alpha),\quad\nu=0,1,\dots,-\alpha-1.$ (2.13)

Due to (2.2)-(2.6), the generalized Laguerre functions satisfy the following
recurrence relations:

######  Lemma 2.5.

For any $k\in\mathbb{N}_{0}$, it holds that

$\displaystyle{\beta}xl_{k}^{\alpha,\beta}(x)=-(k+1)l_{k+1}^{\alpha,\beta}(x)+(2k+\alpha+1)l_{k}^{\alpha,\beta}(x)-(k+\alpha)l_{k-1}^{\alpha,\beta}(x),$
(2.14) $\displaystyle
l^{\alpha,\beta}_{k}(x)=l^{\alpha+1,\beta}_{k}(x)-l^{\alpha+1,\beta}_{k-1}(x),$
(2.15) $\displaystyle\partial_{x}l_{k}^{\alpha,\beta}(x)=-\beta
l_{k-1}^{\alpha+1,\beta}(x)-\frac{\beta}{2}l_{k}^{\alpha,\beta}(x)=-\frac{\beta}{2}\big{[}l_{k}^{\alpha+1,\beta}(x)+l_{k-1}^{\alpha+1,\beta}(x)\big{]},$
(2.16)
$\displaystyle\begin{split}&x\partial_{x}l_{k}^{\alpha,\beta}(x)=\frac{k+1}{2}l_{k+1}^{\alpha,\beta}(x)-\frac{\alpha+1}{2}l_{k}^{\alpha,\beta}(x)-\frac{k+\alpha}{2}l_{k-1}^{\alpha,\beta}(x),\end{split}$
(2.17)
$\displaystyle\begin{split}\partial_{x}l_{k}^{\alpha,\beta}(x)-\partial_{x}l_{k+1}^{\alpha,\beta}(x)=\frac{1}{2}\beta(l_{k}^{\alpha,\beta}(x)+l_{k+1}^{\alpha,\beta}(x)).\end{split}$
(2.18)

Hereafter, we use the convention that $l_{k}^{\alpha,\beta}(x)=0$ whenever
$k\in\mathbb{Z}^{-}$.

The generalized Laguerre functions are eigenfunctions of certain singular
Sturm-Liouville differential operators.

######  Lemma 2.6.

For any $n\in\mathbb{N}_{0}$, it holds that

$\displaystyle\sum_{\nu=0}^{n}(-1)^{\nu}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}x^{-\alpha}\partial_{x}^{\nu}\Big{(}x^{\alpha+n}\partial_{x}^{\nu}l^{\alpha,\beta}_{k}\Big{)}=\frac{\beta^{n}}{2^{n}}\lambda_{k,n}^{\alpha}l^{\alpha,\beta}_{k},\quad
k\geq 0,$ (2.19)

where $\lambda^{\alpha}_{k,n}$ satisfies the following recurrence relation,

$\displaystyle\lambda^{\alpha}_{k,0}=1,\qquad\lambda^{\alpha}_{k,n}=(k+\alpha+1)\lambda^{\alpha+1}_{k,n-1}+k\lambda^{\alpha+1}_{k-1,n-1},\quad
n\geq 1,\;k\geq 0.$ (2.20)

###### Proof.

We prove (2.19) and (2.20) by induction. It is obvious that (2.19) holds for
$n=0$. Moreover, by virtue of (2.7) and (2.11) we have

$\displaystyle-x^{-\alpha}\partial_{x}\Big{(}x^{\alpha+1}\partial_{x}l_{k}^{\alpha,\beta}(x)\Big{)}+\frac{\beta^{2}}{4}xl_{k}^{\alpha,\beta}(x)=\frac{\beta}{2}(2k+\alpha+1)l_{k}^{\alpha,\beta}(x)=\frac{\beta}{2}\lambda^{\alpha}_{k,1}l_{k}^{\alpha,\beta}(x),\quad
k\geq 0,$

which gives (2.19) and (2.20) for $n=1$.

We now assume that (2.19) and (2.20) hold for an integer $n\geq 1$. Then by
the recursive formula of binomial coefficients together with (2.15) and
(2.16),

$\displaystyle I:=$
$\displaystyle\sum_{\nu=0}^{n+1}(-1)^{\nu}\binom{n+1}{\nu}\frac{\beta^{2n+2-2\nu}}{2^{2n+2-2\nu}}x^{-\alpha}\partial_{x}^{\nu}\Big{(}x^{n+1+\alpha}\partial_{x}^{\nu}l^{\alpha,\beta}_{k}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{\nu=0}^{n+1}(-1)^{\nu}\left[\binom{n}{\nu}+\binom{n}{\nu-1}\right]\frac{\beta^{2n+2-2\nu}}{2^{2n+2-2\nu}}x^{-\alpha}\partial_{x}^{\nu}\Big{(}x^{n+1+\alpha}\partial_{x}^{\nu}l^{\alpha,\beta}_{k}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{\beta^{2}}{4}x\sum_{\nu=0}^{n}(-1)^{\nu}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}x^{-(\alpha+1)}\partial_{x}^{\nu}\Big{[}x^{n+(\alpha+1)}\partial_{x}^{\nu}\Big{(}l^{\alpha+1,\beta}_{k}-l^{\alpha+1,\beta}_{k-1}\Big{)}\Big{]}$
$\displaystyle+\frac{\beta}{2}x^{-\alpha}\partial_{x}\sum_{\nu=1}^{n+1}(-1)^{\nu-1}\binom{n}{\nu-1}\frac{\beta^{2n-2(\nu-1)}}{2^{2n-2(\nu-1)}}\partial_{x}^{\nu-1}\Big{[}x^{n+(\alpha+1)}\partial_{x}^{\nu-1}\left(l^{\alpha+1,\beta}_{k}+l^{\alpha+1,\beta}_{k-1}\right)\Big{]}.$

Thus by the induction assumption, (2.14), (2.17) and (2.15), we derive that

$\displaystyle I=$
$\displaystyle\frac{\beta^{n+2}}{2^{n+2}}x\left[\lambda^{\alpha+1}_{k,n}l^{\alpha+1,\beta}_{k}-\lambda^{\alpha+1}_{k-1,n}l^{\alpha+1,\beta}_{k-1}\right]+\frac{\beta^{n+1}}{2^{n+1}}x^{-\alpha}\partial_{x}\left[x^{\alpha+1}\left(\lambda^{\alpha+1}_{k,n}l^{\alpha+1,\beta}_{k}+\lambda^{\alpha+1}_{k-1,n}l^{\alpha+1,\beta}_{k-1}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\beta^{n+1}}{2^{n+1}}\lambda^{\alpha+1}_{k,n}\left[(\alpha+1+\tfrac{\beta}{2}x)l^{\alpha+1,\beta}_{k}+x\partial_{x}l^{\alpha+1,\beta}_{k}\right]+\frac{\beta^{n+1}}{2^{n+1}}\lambda^{\alpha+1}_{k-1,n}\left[(\alpha+1-\tfrac{\beta}{2}x)l^{\alpha+1,\beta}_{k-1}+x\partial_{x}l^{\alpha+1,\beta}_{k-1}\right]$
$\displaystyle=$
$\displaystyle\frac{\beta^{n+1}}{2^{n+1}}\lambda^{\alpha+1}_{k,n}(k+\alpha+1)\big{[}l^{\alpha+1,\beta}_{k}-l^{\alpha+1,\beta}_{k-1}\big{]}+\frac{\beta^{n+1}}{2^{n+1}}\lambda^{\alpha+1}_{k-1,n}k\big{[}l^{\alpha+1,\beta}_{k}-l^{\alpha+1,\beta}_{k-1}\big{]}$
$\displaystyle=$
$\displaystyle\frac{\beta^{n+1}}{2^{n+1}}\big{[}(k+\alpha+1)\lambda^{\alpha+1}_{k,n}+k\lambda^{\alpha+1}_{k-1,n}\big{]}l^{\alpha,\beta}_{k},$

which is exactly (2.19) and (2.20) with $n+1$ in place of $n$. This ends the
proof. ∎

For any $n\in\mathbb{N}_{0}$, $\alpha\in\mathbb{R}$ and $\beta>0$, define the
bilinear form on $H^{n}_{w^{\alpha+n}}(\Lambda)\times
H^{n}_{w^{\alpha+n}}(\Lambda)$,

$\displaystyle
a^{\alpha,\beta}_{n}(u,v)=\sum_{\nu=0}^{n}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}(\partial_{x}^{\nu}u,\partial_{x}^{\nu}v)_{w^{\alpha+n}}.$
(2.21)

It is obvious that $a^{\alpha,\beta}_{n}(\cdot,\cdot)$ is an inner product on
$H^{n}_{w^{\alpha+n}}(\Lambda)$ if $\alpha+n>-1$.

######  Theorem 2.1.

The generalized Laguerre functions
$l_{k}^{\alpha,\beta}(x),\,k\geq\chi(\alpha)$ for $\alpha\in\aleph$ are
mutually orthogonal with respect to the weight function $w{{}^{\alpha}}$,

$a^{\alpha,\beta}_{0}(l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{m})=(l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{m})_{w^{\alpha}}=\beta^{-\alpha-1}\gamma_{k}^{\alpha}\delta_{k,m},\qquad
k,m\geq\chi(\alpha).$ (2.22)

More generally, for any $n\in\mathbb{N}_{0}$ and $\alpha+n\in\aleph$,

$\displaystyle
a^{\alpha,\beta}_{n}(l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{m})=\beta^{n-\alpha-1}\gamma^{\alpha}_{k,n}\delta_{k,m},\qquad
k,m\geq\chi_{n}(\alpha),$ (2.23)

where the positive numbers $\gamma^{\alpha}_{k,n}$ satisfy the recurrence
relation

$\displaystyle\gamma^{\alpha+n}_{k,0}=\gamma^{\alpha+n}_{k},\qquad\gamma^{\alpha}_{k,n}=\frac{1}{2}\big{[}\gamma^{\alpha+1}_{k,n-1}+\gamma^{\alpha+1}_{k-1,n-1}\big{]},\qquad
n\geq 1,$ (2.24)

under the convention that $\gamma^{\alpha}_{k,n}=0$ whenever
$k\in\mathbb{Z}^{-}$.

###### Proof.

The orthogonality (2.22) is an immediate consequence of (2.10). Meanwhile, the
recursive formula (2.21) of binomial coefficients together with (2.15) and
(2.16) yields

$\displaystyle a^{\alpha,\beta}_{n+1}$
$\displaystyle\big{(}l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{m}\big{)}=\sum_{\nu=0}^{n+1}\Big{[}\binom{n}{\nu-1}+\binom{n}{\nu}\Big{]}\frac{\beta^{2n+2-2\nu}}{2^{2n+2-2\nu}}\big{(}\partial_{x}^{\nu}l^{\alpha,\beta}_{k},\partial_{x}^{\nu}l^{\alpha,\beta}_{m}\big{)}_{w^{\alpha+n+1}}$
$\displaystyle=$
$\displaystyle\,\frac{\beta^{2}}{4}\sum_{\nu=1}^{n+1}\binom{n}{\nu-1}\frac{\beta^{2n-2(\nu-1)}}{2^{2n-2(\nu-1)}}\big{(}\partial_{x}^{\nu-1}\big{[}l^{\alpha+1,\beta}_{k}+l^{\alpha+1,\beta}_{k-1}\big{]},\partial_{x}^{\nu-1}\big{[}l^{\alpha+1,\beta}_{m}+l^{\alpha+1,\beta}_{m-1}\big{]}\big{)}_{w^{n+(\alpha+1)}}$
$\displaystyle\,+\frac{\beta^{2}}{4}\sum_{\nu=0}^{n}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}\big{(}\partial_{x}^{\nu}\big{[}l^{\alpha+1,\beta}_{k}-l^{\alpha+1,\beta}_{k-1}\big{]},\partial_{x}^{\nu}\big{[}l^{\alpha+1,\beta}_{m}-l^{\alpha+1,\beta}_{m-1}\big{]}\big{)}_{w^{n+(\alpha+1)}}$
(2.25) $\displaystyle=$
$\displaystyle\,\frac{\beta^{2}}{2}\sum_{\nu=0}^{n}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}\big{[}\big{(}\partial_{x}^{\nu}l^{\alpha+1,\beta}_{k},\partial_{x}^{\nu}l^{\alpha+1,\beta}_{m}\big{)}_{w^{n+(\alpha+1)}}+\big{(}\partial_{x}^{\nu}l^{\alpha+1,\beta}_{k-1},\partial_{x}^{\nu}l^{\alpha+1,\beta}_{m-1}\big{)}_{w^{n+(\alpha+1)}}\big{]}$
$\displaystyle=$
$\displaystyle\,\frac{\beta^{2}}{2}\big{[}a^{\alpha+1,\beta}_{n}\big{(}l^{\alpha+1,\beta}_{k},l^{\alpha+1,\beta}_{m}\big{)}+a^{\alpha+1,\beta}_{n}\big{(}l^{\alpha+1,\beta}_{k-1},l^{\alpha+1,\beta}_{m-1}\big{)}\big{]}.$

To complete the proof of (2.23), we proceed by induction on $n.$ By (2.3) we
get

$\displaystyle
a^{\alpha,\beta}_{1}\big{(}l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{m}\big{)}=\frac{\beta^{2}}{2}\big{[}\beta^{-(\alpha+1)-1}\gamma^{\alpha+1}_{k,0}+\beta^{-(\alpha+1)-1}\gamma^{\alpha+1}_{k-1,0}\big{]}\delta_{k,m}=\frac{\beta^{-\alpha}}{2}\big{[}\gamma^{\alpha+1}_{k,0}+\gamma^{\alpha+1}_{k-1,0}\big{]}\delta_{k,m},$
(2.26)

if either (a). $\alpha+1\in\\{-1,-2,-3,\cdots\\}$ and $k,m\geq-\alpha$; or
(b). $\alpha+1\in(-1,+\infty)$ and $k,m\geq 0.$ This exactly gives (2.23) for
$k,m\geq\chi_{1}(\alpha)$ with $n=1$.

Assume that the result (2.23) for $k,m\geq\chi_{n}(\alpha)$ with $n=p$ holds.
We now verify the result with $n=p+1.$ Clearly, by (2.3) we have

$\displaystyle a^{\alpha,\beta}_{p+1}$
$\displaystyle\big{(}l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{m}\big{)}=\frac{\beta^{2}}{2}\big{[}\beta^{p-(\alpha+1)-1}\gamma^{\alpha+1}_{k,p}+\beta^{p-(\alpha+1)-1}\gamma^{\alpha+1}_{k-1,p}\big{]}\delta_{k,m}=\frac{\beta^{p-\alpha}}{2}\big{[}\gamma^{\alpha+1}_{k,p}+\gamma^{\alpha+1}_{k-1,p}\big{]}\delta_{k,m},$

if either (a). $\alpha+1\in\\{-p-1,-p-2,\cdots\\}$ and $k,m\geq-\alpha$; or
(b). $\alpha+1>-p-1$ and $k,m\geq 0$. This statement implies the result (2.23)
for $k,m\geq\chi_{n+1}(\alpha)$ with $n=p+1.$ This ends the proof. ∎

The normalization constants $\gamma^{\alpha}_{k,n}$ and the eigenvalues
$\lambda^{\alpha}_{k,n}$ are closely related. In effect, for any
$\alpha\in\aleph$, $k\geq\chi(\alpha),\,n\in\mathbb{N}_{0}$, we get that

$\displaystyle\begin{split}\gamma^{\alpha}_{k,n}\overset{\eqref{2.27s}}{=}&\,\beta^{\alpha+1-n}a^{\alpha,\beta}_{n}(l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{k})\overset{\eqref{innProd}}{=}\beta^{\alpha+1-n}\sum_{\nu=0}^{n}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}\big{(}\partial_{x}^{\nu}l^{\alpha,\beta}_{k},\partial_{x}^{\nu}l^{\alpha,\beta}_{k}\big{)}_{w^{\alpha+n}}\\\
=&\,\beta^{\alpha+1-n}\sum_{\nu=0}^{n}(-1)^{\nu}\binom{n}{\nu}\frac{\beta^{2n-2\nu}}{2^{2n-2\nu}}\big{(}\partial_{x}^{\nu}\big{[}w^{\alpha+n}\partial_{x}^{\nu}l^{\alpha,\beta}_{k}\big{]},l^{\alpha,\beta}_{k}\big{)}\\\
\overset{\eqref{2.20s}}{=}&\,\beta^{\alpha+1-n}\frac{\beta^{n}}{2^{n}}\lambda^{\alpha}_{k,n}\big{(}l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{k}\big{)}_{w^{\alpha}}\overset{\eqref{2.26s}}{=}\frac{1}{2^{n}}\lambda^{\alpha}_{k,n}\gamma^{\alpha}_{k},\end{split}$
(2.27)

where the third inequality sign is obtained by integration by parts combined
with (2.13).

Moreover, for sufficiently large $k$, an induction procedure starting with
(2.20) reveals

$\displaystyle\lambda^{\alpha}_{k,n}=(2k+\alpha+1)^{n}+\mathcal{O}(k^{n-2}),$
(2.28)

which implies

$\displaystyle\frac{\gamma^{\alpha}_{k,n}}{\gamma^{\alpha}_{k,n+1}}=\frac{2}{2k+\alpha+1}+\mathcal{O}(k^{-3}).$
(2.29)

The following eigenvalues $\lambda^{-n}_{k,n}$ and normalization constants
$\gamma^{-n}_{k,n}$ are of our particular interest,

$\displaystyle\lambda^{0}_{k,0}=1,\qquad\qquad\quad\lambda^{-n}_{k,n}=2^{n}(k-n+1)_{n},\qquad
k,n\in\mathbb{N}_{0},$ (2.30)
$\displaystyle\gamma^{0}_{k,0}=1,\qquad\qquad\quad\gamma^{-n}_{k,n}=\frac{1}{2^{n}}\sum_{\nu=0}^{\min(k,n)}\binom{n}{\nu},\qquad
k,n\in\mathbb{N}_{0}.$ (2.31)

##  3\. Fully diagonalized spectral methods

In this section, we propose the fully diagonalized spectral methods using
generalized Laguerre functions for solving differential equations on the half
line. The main idea is to find a system of Sobolev orthogonal functions [8,
21] with respect to the coercive bilinear form arising from differential
equation, such that both the exact solution and the approximate solution can
be explicitly expressed as a Fourier series in the Sobolev orthogonal
functions. Although we only consider in this section non-homogenous
Robin/Drichlet boundary value problems of a second order equation, one can
extend the fully diagonalized spectral methods for solving partial
differential equations of an arbitrary high order.

###  3.1. Robin boundary value problems

Consider the second order elliptic boundary value problem:

$\left\\{\begin{array}[]{ll}-u^{\prime\prime}(x)+\gamma
u(x)=f(x),\quad\gamma\geq 0,\quad x\in\Lambda,\\\\[6.0pt] -u^{\prime}(0)+\mu
u(0)=\eta,\quad\displaystyle\lim_{{x\rightarrow+\infty}}u(x)=0,\quad\mu\geq
0.\end{array}\right.$ (3.1)

A weak formulation of (3.1) is to find $u\in H^{1}(\Lambda)$ such that

$A_{\gamma,\mu}(u,v):=\mu
u(0)v(0)+(u^{\prime},v^{\prime})+\gamma(u,v)=(f,v)+\eta\,v(0),\quad\forall
v\in H^{1}(\Lambda).$ (3.2)

The Lax-Milgram lemma guarantees a unique solution to (3.2) if
$f\in(H^{1}(\Lambda))^{\prime}$.

Let

$X_{N}^{\beta}:=\\{e^{-\frac{1}{2}\beta
x}p(x):p\in\mathbb{P}_{N}\\}=\\{l^{-1,\beta}_{k}:0\leq k\leq N\\}.$

The generalized Laguerre spectral scheme for (3.1) is to find $u_{N}\in
X^{\beta}_{N}$, such that

$A_{\gamma,\mu}(u_{N},v_{N})=(f,v_{N})+\eta v_{N}(0),\quad\forall v_{N}\in
X^{\beta}_{N}.$ (3.3)

For an efficient approximation scheme, one usually chooses the generalized
Laguerre functions $\\{l_{k}^{-1,\beta}(x)\\}_{0\leq k\leq N}$ as the basis
functions for problem (3.3). However, we are eager for an ideal approximation
scheme whose (total) stiff matrix, in analogue to the Fourier spectral method
for periodic problem, is diagonal. Obviously, the utilization of the basis
functions $\\{l_{k}^{-1,\beta}(x)\\}_{0\leq k\leq N}$ leads to a tridiagonal
algebraic system. To this end, we shall construct new basis functions
$\\{\mathcal{R}_{k}^{\beta}(x)\\}_{0\leq k\leq N}$ which are mutually
orthogonal with respect to the Sobolev inner product
$A_{\gamma,\mu}(\cdot,\cdot)$ instead of $a_{1}^{-1,\beta}(\cdot,\cdot)$
defined in Theorem 2.1.

######  Lemma 3.1.

Let $\mathcal{R}_{k}^{\beta}\in X^{\beta}_{k},\,k\in\mathbb{N}_{0}$ be the
Sobolev orthogonal Laguerre functions such that
$\mathcal{R}_{k}^{\beta}-l^{-1,\beta}_{k}\in X^{\beta}_{k-1}$ and

$A_{\gamma,\mu}(\mathcal{R}_{k}^{\beta},\mathcal{R}_{m}^{\beta})=\rho_{k}\delta_{k,m},\qquad
k,m\in\mathbb{N}_{0}.$ (3.4)

Then $\mathcal{R}_{k}^{\beta}(x),\,k\in\mathbb{N}_{0}$ satisfy the following
recurrence relation,

$\mathcal{R}_{0}^{\beta}(x)=l_{0}^{-1,\beta}(x),\quad\mathcal{R}_{k}^{\beta}(x)=l_{k}^{-1,\beta}(x)-d_{k-1}\mathcal{R}_{k-1}^{\beta}(x),\quad\forall\
k\geq 1,$ (3.5)

where $\rho_{0}=\mu+\dfrac{\beta}{4}+\dfrac{\gamma}{\beta}$ and

$\displaystyle
d_{k-1}=\frac{\beta}{4\rho_{k-1}}-\frac{\gamma}{\beta\rho_{k-1}},\quad\rho_{k}=-d^{2}_{k-1}\rho_{k-1}+\frac{\beta}{2}+\frac{2\gamma}{\beta},\qquad
k\geq 1.$

###### Proof.

By the orthogonality assumption (3.4) of $\\{\mathcal{R}_{k}^{\beta}\\}$,

$\displaystyle
l_{k}^{-1,\beta}(x)=\mathcal{R}_{k}^{\beta}(x)+\displaystyle\sum_{m=0}^{k-1}\frac{A_{\gamma,\mu}(l_{k}^{-1,\beta},\mathcal{R}_{m}^{\beta})}{\rho_{m}}\mathcal{R}_{m}^{\beta}(x).$

Meanwhile, by (3.2) and (2.21), for any $k>m\geq 0$,

$\displaystyle
A_{\gamma,\mu}(l_{k}^{-1,\beta},\mathcal{R}_{m}^{\beta})=a^{-1,\beta}_{1}(l_{k}^{-1,\beta},\mathcal{R}_{m}^{\beta})+\mu
l_{k}^{-1,\beta}(0)\mathcal{R}_{m}^{\beta}(0)+\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}(l_{k}^{-1,\beta},\mathcal{R}_{m}^{\beta}).$

Both the first and the second terms in the righthand side above are zero due
to the orthogonality relation (2.23) of $\\{l_{k}^{-1,\beta}\\}$ and the
homogeneity boundary condition (2.13) for $l_{k}^{-1,\beta},\,k\geq 1$.
Further by (2.15) and the orthogonality relation (2.22) for
$\\{l^{0,\beta}_{k}\\}$,

$\displaystyle A_{\gamma,\mu}(l_{k}^{-1,\beta},\mathcal{R}_{m}^{\beta})=$
$\displaystyle\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}(l_{k}^{-1,\beta},\mathcal{R}_{m}^{\beta})=\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}(l_{k}^{-1,\beta},l_{k-1}^{-1,\beta})\delta_{m,k-1}$
$\displaystyle=$
$\displaystyle\,\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}(l_{k}^{0,\beta}-l_{k-1}^{0,\beta},l_{k-1}^{0,\beta}-l_{k-2}^{0,\beta})\delta_{m,k-1}$
$\displaystyle=$
$\displaystyle\,\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}(-l_{k-1}^{0,\beta},l_{k-1}^{0,\beta})\delta_{m,k-1}=\,\big{(}\frac{\beta}{4}-\frac{\gamma}{\beta}\big{)}\delta_{m,k-1},\quad
k>m\geq 0,$

which, in return, implies

$\displaystyle
l_{k}^{-1,\beta}(x)=\mathcal{R}_{k}^{\beta}(x)+d_{k-1}\mathcal{R}_{k-1}^{\beta}(x),\quad
d_{k-1}=\frac{\beta}{4\rho_{k-1}}-\frac{\gamma}{\beta\rho_{k-1}},\qquad k\geq
1.$

We now turn to the proof of the recurrence identity for $\rho_{k},k\geq 0$.
Firstly, a direct computation shows

$\displaystyle\rho_{0}=$
$\displaystyle\,A_{\gamma,\mu}(\mathcal{R}_{0}^{\beta},\mathcal{R}_{0}^{\beta})=A_{\gamma,\mu}(l_{0}^{-1,\beta},l_{0}^{-1,\beta})=\mu+\frac{\beta}{4}+\frac{\gamma}{\beta}.$

Further, for $k\geq 1$,

$\displaystyle\beta=$
$\displaystyle\,a^{-1,\beta}_{1}(l_{k}^{-1,\beta},l_{k}^{-1,\beta})=A_{\gamma,\mu}(l_{k}^{-1,\beta},l_{k}^{-1,\beta})+\big{(}\frac{\beta^{2}}{4}-\gamma\big{)}(l_{k}^{-1,\beta},l_{k}^{-1,\beta})$
$\displaystyle=$
$\displaystyle\,A_{\gamma,\mu}(\mathcal{R}_{k}^{\beta}(x)+d_{k-1}\mathcal{R}_{k-1}^{\beta}(x),\mathcal{R}_{k}^{\beta}(x)+d_{k-1}\mathcal{R}_{k-1}^{\beta}(x))+\big{(}\frac{\beta^{2}}{4}-\gamma\big{)}(l_{k}^{0,\beta}-l_{k-1}^{0,\beta},l_{k}^{0,\beta}-l_{k-1}^{0,\beta})$
$\displaystyle=$
$\displaystyle\,\rho_{k}+d^{2}_{k-1}\rho_{k-1}+\big{(}\frac{\beta^{2}}{4}-\gamma\big{)}\frac{2}{\beta},$

where we have used (2.23) and (2.24) for the first equality sign, (3.5) and
(2.15) for the third equality sign, and (3.4) and (2.22) for the fourth
equality sign. The proof is completed. ∎

Obviously, $X^{\beta}_{N}=\\{\mathcal{R}^{\beta}_{k}:0\leq k\leq N\\}$. Thus
the variational forms (3.2) and (3.3) together with the orthogonality of
$\\{\mathcal{R}_{k}^{\beta}\\}$ lead to the following main theorem in this
subsection.

######  Theorem 3.1.

Let $u$ and $u_{N}$ be the solution of (3.1) and (3.3), respectively. Then
both $u$ and $u_{N}$ have the explicit representations in
$\\{\mathcal{R}^{\beta}_{k}\\}$,

$\displaystyle
u(x)=\sum_{k=0}^{\infty}\hat{u}_{k}\mathcal{R}_{k}^{\beta}(x),\qquad
u_{N}(x)=\sum_{k=0}^{N}\hat{u}_{k}\mathcal{R}_{k}^{\beta}(x),$
$\displaystyle\hat{u}_{k}=\frac{1}{\rho_{k}}A_{\gamma,\mu}(u,\mathcal{R}_{k}^{\beta})=\frac{1}{\rho_{k}}\left[(f,\mathcal{R}_{k}^{\beta})+\eta\mathcal{R}_{k}^{\beta}(0)\right],\quad
k\geq 0.$

###  3.2. Dirichlet boundary value problems

Consider the second order elliptic boundary value problem:

$\left\\{\begin{array}[]{ll}-u^{\prime\prime}(x)+\gamma
u(x)=f(x),&\gamma>0,\quad x\in\Lambda,\\\\[6.0pt]
u(0)=\eta,\quad\displaystyle\lim_{{x\rightarrow+\infty}}u(x)=0.\end{array}\right.$
(3.6)

A weak formulation of (3.6) is to find $u\in H^{1}(\Lambda)$ such that
$u(0)=\eta$ and

$A_{\gamma}(u,v):=(u^{\prime},v^{\prime})+\gamma(u,v)=(f,v),\quad\forall v\in
H^{1}_{0}(\Lambda).$ (3.7)

Clearly, if $f\in(H^{1}_{0}(\Lambda))^{\prime}$, then by Lax-Milgram lemma,
(3.7) admits a unique solution.

Let

$X_{N}^{0,\beta}:=\\{e^{-\frac{1}{2}\beta x}p(x):p(0)=0\text{ and
}p\in\mathbb{P}_{N}\\}=\\{l^{-1,\beta}_{k}:1\leq k\leq N\\}.$

The generalized Laguerre spectral scheme for (3.6) is to find $u_{N}\in
X^{\beta}_{N}$, such that $u_{N}(0)=\eta$ and

$A_{\gamma}(u_{N},v_{N})=(f,v_{N}),\quad\forall v_{N}\in X^{0,\beta}_{N}.$
(3.8)

To propose a fully diagonal approximation scheme for (3.7) , we need to
construct new basis functions $\\{\mathcal{S}_{k}^{\beta}\\}_{1\leq k\leq N}$
which are mutually orthogonal with respect to the Sobolev inner product
$A_{\gamma}(\cdot,\cdot)$.

######  Lemma 3.2.

Let $\mathcal{S}_{k}^{\beta}\in X^{0,\beta}_{k},\,k\geq 1$ be the Sobolev
orthogonal Laguerre functions such that
$\mathcal{S}_{k}^{\beta}-l^{-1,\beta}_{k}\in X^{0,\beta}_{k-1}$ and

$A_{\gamma}(\mathcal{S}_{k}^{\beta},\mathcal{S}_{m}^{\beta})=\varrho_{k}\delta_{k,m},\quad
k,m\geq 1.$ (3.9)

Then we have

$\mathcal{S}_{1}^{\beta}(x)=l_{1}^{-1,\beta}(x),\quad\mathcal{S}_{k}^{\beta}(x)=l_{k}^{-1,\beta}(x)-d_{k-1}\mathcal{S}_{k-1}^{\beta}(x),\quad
k\geq 2,$ (3.10)

where $\varrho_{1}=\dfrac{4\gamma+\beta^{2}}{2\beta}$ and

$\displaystyle
d_{k-1}=\frac{\beta}{4\varrho_{k-1}}-\frac{\gamma}{\beta\varrho_{k-1}},\quad\varrho_{k}=-d^{2}_{k-1}\varrho_{k-1}+\frac{\beta}{2}+\frac{2\gamma}{\beta},\qquad
k\geq 2.$

###### Proof.

The proof is in the same way as Lemma 3.1. We neglect the details. ∎

To deal with the non-homogenous boundary condition, we need to supplement
$\mathcal{S}_{0}^{\beta}(x)$ which is orthogonal to all functions in
$H^{1}_{0}(\Lambda)$ with respect to $A_{\gamma}(\cdot,\cdot)$. Suppose

$\displaystyle\mathcal{S}_{0}^{\beta}(x)=\sum_{m=0}^{\infty}\hat{s}_{m}l_{m}^{-1,\beta}(x)\quad\text{
with }\hat{s}_{0}=1,$

such that $\mathcal{S}_{0}^{\beta}(0)=1$. Then by (2.15), (2.23) and (2.22),

$\displaystyle 0=$
$\displaystyle\,A_{\gamma}(\mathcal{S}_{0}^{\beta},l_{k}^{-1,\beta})=a_{1}^{\alpha,\beta}(\mathcal{S}_{0}^{\beta},l_{k}^{-1,\beta})+\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}(\mathcal{S}_{0}^{\beta},l_{k}^{-1,\beta})$
$\displaystyle=$
$\displaystyle\,\displaystyle\sum_{m=0}^{\infty}\hat{s}_{m}a_{1}^{\alpha,\beta}(l_{m}^{-1,\beta},l_{k}^{-1,\beta})+\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}\displaystyle\sum_{m=0}^{\infty}\hat{s}_{m}\big{(}l_{m}^{0,\beta}-l_{m-1}^{0,\beta},l_{k}^{0,\beta}-l_{k-1}^{0,\beta}\big{)}$
$\displaystyle=$
$\displaystyle\,\beta\hat{s}_{k}+\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}\frac{2\hat{s}_{k}-\hat{s}_{k-1}-\hat{s}_{k+1}}{\beta}$
$\displaystyle=$
$\displaystyle\,\big{(}\gamma+\frac{\beta^{2}}{4}\big{)}\frac{2\hat{s}_{k}}{\beta}-\big{(}\gamma-\frac{\beta^{2}}{4}\big{)}\frac{\hat{s}_{k-1}+\hat{s}_{k+1}}{\beta},\quad
k\geq 1.$

The characteristic equation for the above three term recurrence relation reads

$\displaystyle(\beta^{2}-4\gamma\big{)}z^{2}+2\big{(}4\gamma+\beta^{2}\big{)}z+(\beta^{2}-4\gamma\big{)}=0,$

which admits two distinct real roots
$z_{\pm}=\displaystyle\frac{2\sqrt{\gamma}\mp\beta}{2\sqrt{\gamma}\pm\beta}$
if and only if $\gamma>0$. In this case, all $\hat{s}_{m}$, $m\geq 0$ can be
expressed as

$\displaystyle\hat{s}_{m}=c_{+}\frac{(2\sqrt{\gamma}-\beta)^{m}}{(2\sqrt{\gamma}+\beta)^{m}}+c_{-}\frac{(2\sqrt{\gamma}+\beta)^{m}}{(2\sqrt{\gamma}-\beta)^{m}},$
(3.11)

with the coefficients $c_{\pm}$ to be determined by $\hat{s}_{0}=1$ and
$\lim_{m\to\infty}\hat{s}_{m}=0$. As a result,

$\displaystyle c_{+}=1,\quad c_{-}=0,\quad\text{ and
}\quad\mathcal{S}_{0}^{\beta}(x)=\sum_{m=0}^{\infty}\dfrac{(2\sqrt{\gamma}-\beta)^{m}}{(2\sqrt{\gamma}+\beta)^{m}}l_{m}^{-1,\beta}(x).$

Also, we need a function $\mathcal{S}^{\beta}_{0,N}\in X^{\beta}_{N}$ which
satisfies $\mathcal{S}^{\beta}_{0,N}(0)=1$ and is orthogonal to all functions
in $X^{0,\beta}_{N}$ with respect to $A_{\gamma}(\cdot,\cdot)$. Let us write

$\displaystyle\mathcal{S}_{0,N}^{\beta}(x)=\sum_{m=0}^{N}\hat{s}_{m}l_{k}^{-1,\beta}(x).$

Then $\\{\hat{s}_{m}\\}_{0\leq m\leq N}$ is determined by (3.11) for $1\leq
m\leq N$ together with the endpoint values $\hat{s}_{0}=1$ and
$\hat{s}_{N+1}=0$. Solving the system for $c_{\pm}$ we finally have

$\displaystyle
c_{+}=\frac{(2\sqrt{\gamma}+\beta)^{2N+2}}{(2\sqrt{\gamma}+\beta)^{2N+2}-(2\sqrt{\gamma}-\beta)^{2N+2}},\qquad
c_{-}=-\frac{(2\sqrt{\gamma}-\beta)^{2N+2}}{(2\sqrt{\gamma}+\beta)^{2N+2}-(2\sqrt{\gamma}-\beta)^{2N+2}},$
$\displaystyle\mathcal{S}_{0,N}^{\beta}(x)=\sum_{m=0}^{N}\frac{(2\sqrt{\gamma}+\beta)^{2N+2-m}(2\sqrt{\gamma}-\beta)^{m}-(2\sqrt{\gamma}-\beta)^{2N+2-m}(2\sqrt{\gamma}+\beta)^{m}}{(2\sqrt{\gamma}+\beta)^{2N+2}-(2\sqrt{\gamma}-\beta)^{2N+2}}l_{k}^{-1,\beta}(x).$

######  Theorem 3.2.

Let $u$ and $u_{N}$ be the solution to (3.6) and (3.8), respectively. Then
both $u$ and $u_{N}$ have the explicit representations in
$\\{\mathcal{S}^{\beta}_{k}\\}$,

$\displaystyle
u(x)=\eta\mathcal{S}_{0}^{\beta}(x)+\sum_{k=1}^{\infty}\hat{u}_{k}\mathcal{S}_{k}^{\beta}(x),\qquad
u_{N}(x)=\eta\mathcal{S}_{0,N}^{\beta}(x)+\sum_{k=1}^{N}\hat{u}_{k}\mathcal{S}_{k}^{\beta}(x),$
$\displaystyle\hat{u}_{k}=\frac{1}{\varrho_{k}}A_{\gamma}(u,\mathcal{S}_{k}^{\beta})=\frac{(f,\mathcal{S}^{\beta}_{m})}{\varrho_{m}},\quad
k\geq 1.$

##  4\. Convergence analysis

In this section, we shall derive the optimal error estimate for the spectral
methods using generalized Laguerre functions. To this end, we first conduct
some numerical analysis on the orthogonal projections.

###  4.1. Orthogonal projections

Let $r,N\in\mathbb{N}_{0}$, $\alpha>-r-1$ and $\beta>0$. Define the orthogonal
projection $\pi^{\alpha,\beta}_{r,N}:H^{r}_{w^{\alpha+r}}(\Lambda)\mapsto
X^{\beta}_{N}$ such that

$\displaystyle a^{\alpha,\beta}_{r}(u-\pi^{\alpha,\beta}_{r,N}u,v)=0,\quad
v\in X^{\beta}_{N}.$ (4.1)

In view of the orthogonality relation (2.23), $\pi^{\alpha,\beta}_{r,N}u$ is a
truncated Fourier series of $u$ in $\\{l^{\alpha,\beta}_{k}\\}$,

$\displaystyle\pi^{\alpha,\beta}_{r,N}u(x)=\sum_{k=0}^{N}\hat{u}_{k}l^{\alpha,\beta}_{k}(x),\quad
u(x)=\sum_{k=0}^{\infty}\hat{u}_{k}l^{\alpha,\beta}_{k}(x),\qquad\hat{u}_{k}=\frac{a^{\alpha,\beta}_{r}(u,l^{\alpha,\beta}_{k})}{a^{\alpha,\beta}_{r}(l^{\alpha,\beta}_{k},l^{\alpha,\beta}_{k})}.$
(4.2)

In return, one obtains that, for any nonnegative integers $N$ and $s$ with
$N,s>-\alpha-1$,

$\displaystyle a^{\alpha,\beta}_{s}(u-\pi^{\alpha,\beta}_{r,N}u,v)=0,\quad
v\in X^{\beta}_{N},$

which states that $\pi^{\alpha,\beta}_{r,N}=\pi^{\alpha,\beta}_{s,N}$ for all
admissible $s,r$ and $N$. For this reason, we shall omit the subscript $r$ and
simplify write $\pi^{\alpha,\beta}_{N}=\pi^{\alpha,\beta}_{r,N}$.

Besides, (2.12) clearly states $\partial_{x}^{\ell}l^{-n,\beta}_{k}(0)=0$,
$\ell=0,1,\dots,n-1$ for any $k\geq n\geq 1$. Thus for $N\geq n$,
$\pi_{N}^{-n,\beta}u$ preserves the endpoint values of $u$ up to the
$(n-1)$-th order derivative, i.e.,

$\displaystyle\partial_{x}^{\ell}\pi_{N}^{-n,\beta}u(0)=\partial_{x}^{\ell}u(0),\qquad\ell=0,1,\dots,n-1.$
(4.3)

In other words, $u-\pi_{N}^{-n,\beta}u\in H^{n}_{0}(\Lambda)$ for any $u\in
H^{n}(\Lambda)$ if $N\geq n$.

To measure the error between $u$ and $\pi_{N}^{\alpha,\beta}u$, we introduce
the equivalent norm in $H^{r}_{w^{\alpha+r}}(\Lambda)$ for
$r\in\mathbb{N}_{0}$ and $\alpha\in\mathbb{R}$,

$\displaystyle\|u\|_{r,\alpha,\beta}=\big{[}a^{\alpha,\beta}_{r}(u,u)\big{]}^{1/2}=\Big{[}\sum_{\nu=0}^{r}\binom{r}{\nu}\frac{\beta^{2r-2\nu}}{2^{2r-2\nu}}\big{(}\partial_{x}^{\nu}u,\partial_{x}^{\nu}u\big{)}_{w^{\alpha+r}}\Big{]}^{1/2}.$

######  Theorem 4.1.

Let $r\in\mathbb{N}_{0}$, $\alpha>-r-1$ and $\beta>0$. Then for any function
$u\in H^{r}_{w^{\alpha+r}}(\Lambda)$ and any nonnegative integers
$N,s>-\alpha-1$,

$\displaystyle\|u-\pi_{N}^{\alpha,\beta}u\|_{s,\alpha,\beta}\lesssim(\beta
N)^{\frac{s-r}{2}}\|u-\pi_{N}^{\alpha,\beta}u\|_{r,\alpha,\beta}\lesssim(\beta
N)^{\frac{s-r}{2}}\|u\|_{r,\alpha,\beta},\quad r\geq s,$ (4.4)

where the implicit constants $c=c(\alpha,r)$ is independent of $\beta$, $N$,
$s$ and $u$.

###### Proof.

By (4.2) and the orthogonality (2.23),

$\displaystyle\|u-\pi^{\alpha,\beta}_{N}$ $\displaystyle
u\|_{s,\alpha,\beta}^{2}=\sum_{k=N+1}^{\infty}\beta^{s-\alpha-1}\gamma^{\alpha}_{k,s}\hat{u}_{k}^{2}=\beta^{s-r}\sum_{k=N+1}^{\infty}\left(\frac{\gamma^{\alpha}_{k,s}}{\gamma^{\alpha}_{k,r}}\right)\beta^{r-\alpha-1}\gamma^{\alpha}_{k,r}\hat{u}_{k}^{2}.$

By (2.29), one reveals that

$\displaystyle\frac{\gamma^{\alpha}_{k,s}}{\gamma^{\alpha}_{k,r}}\lesssim\frac{2^{r-s}}{(2k+\alpha+1)^{r-s}}\lesssim\frac{1}{N^{r-s}},\qquad
k\geq N+1,\,r\geq s.$

As a result,

$\displaystyle\|u-$
$\displaystyle\pi^{\alpha,\beta}_{N}u\|_{s,\alpha,\beta}^{2}\lesssim(\beta
N)^{s-r}\sum_{k=N+1}^{\infty}\beta^{r-\alpha-1}\gamma^{\alpha}_{k,r}\hat{u}_{k}^{2}$
$\displaystyle=(\beta
N)^{s-r}\|u-\pi^{\alpha,\beta}_{N}u\|_{r,\alpha,\beta}^{2}\leq(\beta
N)^{s-r}\|u\|_{r,\alpha,\beta}^{2},$

which leads to (4.4). ∎

###  4.2. Convergence analysis

We first give the error estimate of the generalized Laguerre spectral method
(3.8) for the non-homogenous Dirichlet boundary value problem (3.6).

######  Theorem 4.2.

Let $u$ and $u_{N}$ be the solutions to (3.6) and (3.8), respectively. If
$u\in H^{r}_{w^{r-1}}(\Lambda)$ and integer $r\geq 1$, then for sufficiently
large $N$,

$\|(u-u_{N})^{\prime}\|^{2}+\gamma\|u-u_{N}\|^{2}\lesssim\Big{(}1+\frac{4\gamma}{\beta^{2}}\Big{)}\,(\beta
N)^{1-r}\|u\|^{2}_{r,-1,\beta},$ (4.5)

and

$\|u-u_{N}\|_{w^{-1}}\lesssim\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}\,(\beta
N)^{-\frac{r}{2}}\|u\|_{r,-1,\beta}.$ (4.6)

###### Proof.

We first prove the inequality (4.5). By (3.7), we get

$\displaystyle((\pi_{N}^{-1,\beta}$ $\displaystyle
u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u,v_{N})=(f,v_{N})+((\pi_{N}^{-1,\beta}u-u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u-u,v_{N}),\quad
v_{N}\in X^{0,\beta}_{N}.$

Subtracting the above equation from (3.8) yields

$\displaystyle((\pi^{-1,\beta}_{N}$ $\displaystyle
u-u_{N})^{\prime},v_{N}^{\prime})+\gamma(\pi^{-1,\beta}_{N}u-u_{N},v_{N})=((\pi_{N}^{-1,\beta}u-u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u-u,v_{N}).$

The above with (4.1) and the Cauchy-Schwartz inequality gives that for any
real number $q$,

$\displaystyle\,\big{|}((\pi^{-1,\beta}_{N}u-u_{N})^{\prime},v_{N}^{\prime})+\gamma(\pi^{-1,\beta}_{N}u-u_{N},v_{N})\big{|}$
$\displaystyle=$
$\displaystyle\,\big{|}(1+q)((\pi_{N}^{-1,\beta}u-u)^{\prime},v_{N}^{\prime})+\frac{4\gamma+\beta^{2}q}{4}(\pi_{N}^{-1,\beta}u-u,v_{N})\big{|}$
$\displaystyle\leq$
$\displaystyle\,|1+q|\,\big{\|}(\pi_{N}^{-1,\beta}u-u)^{\prime}\big{\|}\,\big{\|}v_{N}^{\prime}\big{\|}+\frac{|4\gamma+\beta^{2}q|}{4}\big{\|}\pi_{N}^{-1,\beta}u-u\big{\|}\,\big{\|}v_{N}\big{\|}$
$\displaystyle\leq$
$\displaystyle\,\Big{[}(1+q)^{2}\big{\|}(\pi_{N}^{-1,\beta}u-u)^{\prime}\big{\|}^{2}+\frac{(4\gamma+\beta^{2}q)^{2}}{16\gamma}\big{\|}\pi_{N}^{-1,\beta}u-u\big{\|}^{2}\Big{]}^{1/2}\,\big{[}\|v_{N}^{\prime}\|^{2}+\gamma\|v_{N}\|^{2}\big{]}^{1/2}.$

Taking $v_{N}=u_{N}-\pi_{N}^{-1,\beta}u\in X^{0,\beta}_{N}$, we obtain

$\displaystyle\|(u_{N}-$
$\displaystyle\pi^{-1,\beta}_{N}u)^{\prime}\|^{2}+\gamma\|u_{N}-\pi^{-1,\beta}_{N}u\|^{2}\leq(1+q)^{2}\big{\|}(\pi_{N}^{-1,\beta}u-u)^{\prime}\big{\|}^{2}+\frac{(4\gamma+\beta^{2}q)^{2}}{16\gamma}\big{\|}\pi_{N}^{-1,\beta}u-u\big{\|}^{2}.$

This, along with the triangle inequality, leads to

$\displaystyle\|(u-u_{N})^{\prime}\|^{2}+\gamma\|u-u_{N}\|^{2}$
$\displaystyle\leq
2\|(u-\pi^{-1,\beta}_{N}u)^{\prime}\|^{2}+2\gamma\|u-\pi^{-1,\beta}_{N}u\|^{2}+2\|(u_{N}-\pi^{-1,\beta}_{N}u)^{\prime}\|^{2}+2\gamma\|u_{N}-\pi^{-1,\beta}_{N}u\|^{2}$
$\displaystyle\leq
2(q^{2}+2q+2)\big{\|}(\pi_{N}^{-1,\beta}u-u)^{\prime}\big{\|}^{2}+\frac{\beta^{4}q^{2}+8\beta^{2}\gamma
q+32\gamma^{2}}{8\gamma}\big{\|}\pi_{N}^{-1,\beta}u-u\big{\|}^{2}.$

Now taking $q=-\frac{2\sqrt{2\gamma}}{\beta}$ and using (4.4), we obtain

$\displaystyle\|(u-u_{N}$
$\displaystyle)^{\prime}\|^{2}+\gamma\|u-u_{N}\|^{2}\leq\big{[}2+2\big{(}1-\tfrac{2\sqrt{2\gamma}}{\beta}\big{)}^{2}\big{]}\,\Big{[}\big{\|}(\pi_{N}^{-1,\beta}u-u)^{\prime}\big{\|}^{2}+\frac{\beta^{2}}{4}\big{\|}\pi_{N}^{-1,\beta}u-u\big{\|}^{2}\Big{]}.$
$\displaystyle\lesssim$
$\displaystyle\,\big{(}1+4\gamma/\beta^{2}\big{)}\,(\beta
N)^{1-r}\|u\|^{2}_{r,-1,\beta}.$

We next verify the inequality (4.6) using a duality argument. Consider the
auxiliary problem

$-xv^{\prime\prime}(x)+\gamma xv(x)=u(x)-u_{N}(x)~{}\text{ in
}~{}\Lambda,\qquad v(0)=0.$ (4.7)

Its weak form is

$(\varphi^{\prime},v^{\prime})+\gamma(\varphi,v)=(\varphi,u-u_{N})_{w^{-1}},\quad\forall\varphi\in
H^{1}_{0}(\Lambda),$

which admits a unique solution $v\in H_{0}^{1}(\Lambda)$. Moreover, (4.7)
yields

$\displaystyle\|u-u_{N}\|_{w^{-1}}^{2}=(-xv^{\prime\prime}+\gamma
xv,-v^{\prime\prime}+\gamma
v)=\|v^{\prime\prime}\|_{w}^{2}+\gamma^{2}\|v\|_{w}^{2}+2\gamma\|v^{\prime}\|_{w}^{2}=\|v\|^{2}_{2,-1,2\sqrt{\gamma}},$
(4.8)

where the second equality sign is derived by integration by parts. Further, a
direct computation leads to

$\displaystyle\|v\|_{2,-1,\beta}\leq\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}\|v\|_{2,-1,2\sqrt{\gamma}}.$
(4.9)

Hence, taking $\varphi=u-u_{N}$ and using the Cauchy-Schwartz inequality, we
have

$\displaystyle\|u-u_{N}\|_{w^{-1}}^{2}$
$\displaystyle=(u-u_{N},u-u_{N})_{w^{-1}}=((u-u_{N})^{\prime},v^{\prime})+\gamma(u-u_{N},v)$
$\displaystyle\overset{\eqref{4.3s}}{=}((u-u_{N})^{\prime},(v-\pi^{-1,\beta}_{N}v)^{\prime})+\gamma(u-u_{N},v-\pi^{-1,\beta}_{N}v)$
$\displaystyle\leq\Big{[}\|(u-u_{N})^{\prime}\|^{2}+\frac{4\gamma^{2}}{\beta^{2}}\|u-u_{N}\|^{2}\Big{]}^{\frac{1}{2}}\,\Big{[}\|(v-\pi^{-1,\beta}_{N}v)^{\prime}\|^{2}+\frac{\beta^{2}}{4}\|v-\pi^{-1,\beta}_{N}v\|^{2}\Big{]}^{\frac{1}{2}}$
$\displaystyle\overset{\eqref{errorProj}}{\lesssim}(\beta
N)^{-1/2}\|v\|_{2,-1,\beta}\Big{[}\|(u-u_{N})^{\prime}\|^{2}+\frac{4\gamma^{2}}{\beta^{2}}\|u-u_{N}\|^{2}\Big{]}^{\frac{1}{2}}$
(4.10)
$\displaystyle\overset{\eqref{tran}}{\lesssim}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}(\beta
N)^{-1/2}\|v\|_{2,-1,2\sqrt{\gamma}}\times\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}\Big{[}\|(u-u_{N})^{\prime}\|^{2}+\gamma\|u-u_{N}\|^{2}\Big{]}^{\frac{1}{2}}$
$\displaystyle\overset{\eqref{stab}}{\lesssim}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}(\beta
N)^{-1/2}\|u-u_{N}\|_{w^{-1}}\times\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}(\beta
N)^{1/2-r/2}\|u\|_{r,-1,\beta}$
$\displaystyle\overset{\eqref{4.4s}}{\lesssim}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}(\beta
N)^{-\frac{r}{2}}\|u-u_{N}\|_{w^{-1}}\|u\|_{r,-1,\beta},$

which ends the proof of (4.6). ∎

Next, we present the main theorem on the generalized Laguerre spectral method
for the Robin boundary value problem (3.1).

######  Theorem 4.3.

Let $u$ and $u_{N}$ be the solutions to (3.1) and (3.3), respectively. If
$u\in H^{r}_{w^{r-1}}(\Lambda)$ and integer $r\geq 1$, then for sufficiently
large $N$,

$\mu\big{|}u(0)-u_{N}(0)\big{|}^{2}+\|(u-u_{N})^{\prime}\|^{2}+\gamma\|u-u_{N}\|^{2}\lesssim\Big{(}1+\frac{4\gamma}{\beta^{2}}\Big{)}\,(\beta
N)^{1-r}\|u\|^{2}_{r,-1,\beta},$ (4.11)

and

$\|u-u_{N}\|_{(1+w)^{-1}}\lesssim\Big{(}1+\frac{1}{\sqrt{2\mu}}\Big{)}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}\,(\beta
N)^{-\frac{r}{2}}\|u\|_{r,-1,\beta}.$ (4.12)

###### Proof.

By (4.1) and (3.2),

$\displaystyle\mu$
$\displaystyle\pi_{N}^{-1,\beta}u(0)v_{N}(0)+((\pi_{N}^{-1,\beta}u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u,v_{N})$
$\displaystyle=\mu
u(0)v_{N}(0)+(u^{\prime},v_{N}^{\prime})+\gamma(u,v_{N})+((\pi_{N}^{-1,\beta}u-u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u-u,v_{N})$
$\displaystyle=(f,v_{N})+\eta
v_{N}(0)+((\pi_{N}^{-1,\beta}u-u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u-u,v_{N}),\quad
v_{N}\in X_{N}^{\beta}.$

Substracting the above equation by (3.3) yields

$\displaystyle\mu(\pi_{N}^{-1,\beta}$ $\displaystyle
u(0)-u_{N}(0))v_{N}(0)+((\pi_{N}^{-1,\beta}u-u_{N})^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u-u_{N},v_{N})$
$\displaystyle=$
$\displaystyle\,((\pi_{N}^{-1,\beta}u-u)^{\prime},v_{N}^{\prime})+\gamma(\pi_{N}^{-1,\beta}u-u,v_{N}).$

Then a similar argument as in the proof of (4.5) gives (4.11).

We next verify the inequality (4.12) using a duality argument. For any $g\in
L^{2}_{w^{-1}+w^{-2}}(\Lambda)$, we consider the auxiliary problem

$-xv^{\prime\prime}(x)+\gamma xv(x)=g(x)~{}\text{ in
}~{}\Lambda,\qquad-v^{\prime}(0)+\mu v(0)=0.$ (4.13)

Its weak form is

$\mu\varphi(0)v(0)+(\varphi^{\prime},v^{\prime})+\gamma(\varphi,v)=(\varphi,g)_{w^{-1}},\quad\forall\varphi\in
H^{1}(\Lambda),$

which admits a unique solution $v\in H^{1}(\Lambda)$. Taking
$\varphi=u-u_{N}$, we have

$\mu(u(0)-u_{N}(0))v(0)+(u^{\prime}-u_{N}^{\prime},v^{\prime})+\gamma(u-u_{N},v)=(u-u_{N},g)_{w^{-1}}.$
(4.14)

Moreover, by (3.2) and (3.3) we get

$\mu(u(0)-u_{N}(0))v_{N}(0)+(u^{\prime}-u_{N}^{\prime},v_{N}^{\prime})+\gamma(u-u_{N},v_{N})=0,\quad\forall
v_{N}\in X^{\beta}_{N}.$ (4.15)

Let $v_{N}=\pi_{N}^{-1,\beta}v$ in (4.15). Then by (4.14), (4.15) and (4.3) we
deduce

$((u-u_{N})^{\prime},(v-\pi_{N}^{-1,\beta}v)^{\prime})+\gamma(u-u_{N},v-\pi_{N}^{-1,\beta}v)=(u-u_{N},g)_{w^{-1}}.$
(4.16)

Further by integration by parts, (4.13) yields

$\displaystyle\|g\|_{w^{-2}}^{2}$ $\displaystyle=(-v^{\prime\prime}+\gamma
v,-v^{\prime\prime}+\gamma
v)=\|v^{\prime\prime}\|^{2}+\gamma^{2}\|v\|^{2}+2\gamma\|v^{\prime}\|^{2}+2\gamma\mu
v^{2}(0),$ (4.17)

and

$\displaystyle\|g\|_{w^{-1}}^{2}$ $\displaystyle=(-xv^{\prime\prime}+\gamma
xv,-v^{\prime\prime}+\gamma
v)=\|v^{\prime\prime}\|^{2}_{w}+\gamma^{2}\|v\|^{2}_{w}+2\gamma\|v^{\prime}\|^{2}_{w}-\gamma
v^{2}(0).$ (4.18)

Hence

$\displaystyle\|v\|^{2}_{2,-1,2\sqrt{\gamma}}=\|v^{\prime\prime}\|^{2}_{w}+\gamma^{2}\|v\|^{2}_{w}+2\gamma\|v^{\prime}\|^{2}_{w}\leq\|g\|_{w^{-1}}^{2}+\frac{1}{2\mu}\|g\|_{w^{-2}}^{2}\leq(1+\frac{1}{2\mu})\|g\|_{w^{-1}+w^{-2}}^{2}.$
(4.19)

Thus, a similar argument as (4.2) gives

$\displaystyle|(u-u_{N},g)_{w^{-1}}|$
$\displaystyle\overset{\eqref{sr3}}{=}|((u-u_{N})^{\prime},(v-\pi^{-1,\beta}_{N}v)^{\prime})+\gamma(u-u_{N},v-\pi^{-1,\beta}_{N}v)|$
$\displaystyle\overset{\eqref{errs}}{\lesssim}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}(\beta
N)^{-1/2}\|v\|_{2,-1,2\sqrt{\gamma}}\times\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}\Big{[}\|(u-u_{N})^{\prime}\|^{2}+\gamma\|u-u_{N}\|^{2}\Big{]}^{\frac{1}{2}}$
$\displaystyle\overset{\eqref{stabr2}}{\lesssim}\Big{(}1+\frac{1}{\sqrt{2\mu}}\Big{)}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}(\beta
N)^{-1/2}\|g\|_{w^{-1}+w^{-2}}\times\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}(\beta
N)^{1/2-r/2}\|u\|_{r,-1,\beta}$
$\displaystyle\lesssim\Big{(}1+\frac{1}{\sqrt{2\mu}}\Big{)}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}(\beta
N)^{-\frac{r}{2}}\|g\|_{w^{-1}+w^{-2}}\|u\|_{r,-1,\beta}.$

Finally, we obtain

$\displaystyle\|u-u_{N}\|_{(1+w)^{-1}}=\|\frac{x}{1+x}(u-u_{N})\|_{w^{-1}+w^{-2}}=\sup_{g\in
L^{2}_{w^{-1}+w^{-2}}(\Lambda),~{}g\neq
0}\frac{|(\frac{x}{1+x}(u-u_{N}),g)_{w^{-1}+w^{-2}}|}{\|g\|_{w^{-1}+w^{-2}}}$
$\displaystyle=\sup_{g\in L^{2}_{w^{-1}+w^{-2}}(\Lambda),~{}g\neq
0}\frac{|(u-u_{N},g)_{w^{-1}}|}{\|g\|_{w^{-1}+w^{-2}}}\lesssim\Big{(}1+\frac{1}{\sqrt{2\mu}}\Big{)}\Big{(}1+\frac{\beta^{2}}{4\gamma}\Big{)}\Big{(}1+\frac{2\sqrt{\gamma}}{\beta}\Big{)}^{2}(\beta
N)^{-\frac{r}{2}}\|u\|_{r,-1,\beta},$

which ends the proof of (4.12). ∎

##  5\. Numerical experiments

In this section, we examine the effectiveness and the accuracy of the fully
diagonalized Laguerre spectral method for solving second order elliptic
equations on the half line. The righthand terms
$\\{(f,\mathcal{R}_{k}^{\beta})\\}_{k=0}^{N}$ or
$\\{(f,\mathcal{S}_{k}^{\beta})\\}_{k=1}^{N},$ as well as the discrete errors,
are evaluated through the Laguerre-Gauss quadrature with $2N+1$ nodes (cf.
[30]).

###  5.1. Dirichlet boundary value problems

We first examine the fully diagonalized Laguerre spectral method for the
Dirichlet boundary value problem (3.6). We take $\gamma=1$, $\eta=1$ in (3.6)
and consider the following three cases of the smooth solutions with different
decay properties.

  * •

$u(x)=e^{-x}(\sin(2x)+\eta),$ which is exponential decay with oscillation. In
Figures 5.2, 5.2 and 5.4, we plot the $\log_{10}$ of the discrete $L^{2}$-,
$H^{1}$\- and $L^{2}_{w^{-1}}$-errors vs. $N$ with various values of $\beta$.
Clearly, the approximate solutions converge at exponential rates. We also see
that for fixed $N$, the scheme with $\beta=4$ produces better numerical
results than that with $\beta=1,~{}2.$ However, the choice of the optimal
scaling factor $\beta$ for a given differential equation is still an open
problem. Generally speaking, the choice of $\beta$ depends on the asymptotic
behavior of solutions.

  * •

$u(x)=(x+\eta)(1+x)^{-h}$ with $h>1,$ which is algebraic decay. In Figures
5.4, 5.6 and 5.6, we plot the $\log_{10}$ of the discrete $L^{2}$-, $H^{1}$\-
and $L^{2}_{w^{-1}}$-errors vs. $\log_{10}(\beta N)$ with fixed $\beta=4$ and
various values of $h$. They show that the faster the exact solution decays,
the smaller the numerical errors would be. Clearly,

$\displaystyle\|(x+\eta)(1+x)^{-h}\|_{r,-1,\beta}<\infty,\quad\forall 0\leq
r<2h-2.$

According to Theorem 4.2, the expected $L^{2}$\- and $H^{1}$-errors can be
bounded by $c(\beta N)^{3/2-h+\varepsilon}$ for any $\varepsilon>0$, and the
expected $L^{2}_{w^{-1}}$-error can be bounded by $c(\beta
N)^{1-h+\varepsilon}$ for any $\varepsilon>0$. The observed convergence rates
plotted in Figures 5.4, 5.6 and 5.6 agree well with the theoretical results.

  * •

$u(x)=(\sin(2x)+\eta)(1+x)^{-h}$, which is algebraic decay with oscillation.
In Figures 5.8, 5.8 and 5.10, we plot the $\log_{10}$ of the discrete
$L^{2}$-, $H^{1}$\- and $L^{2}_{w^{-1}}$-errors vs. $\log_{10}(\beta N)$ with
fixed $\beta=4$ and various values of $h$. Since

$\displaystyle\|(\sin(2x)+\eta)(1+x)^{-h}\|_{r,-1,\beta}<\infty,\quad\forall
0\leq r<2h,$

the expected $L^{2}$-, $H^{1}$\- (resp. $L^{2}_{w^{-1}}$-) errors given by
Theorem 4.2 can be bounded by $c(\beta N)^{1/2-h+\varepsilon}$ (resp. $c(\beta
N)^{-h+\varepsilon}$) for any $\varepsilon>0$. The observed convergence rates
plotted in Figures 5.8, 5.8 and 5.10 also agree well with the theoretical
results.

Figure 5.1. $L^{2}$-errors.

Figure 5.2. $H^{1}$-errors.

Figure 5.3. $L^{2}_{w^{-1}}$-errors.

Figure 5.4. $L^{2}$-errors with $\beta=4$.

Figure 5.5. $H^{1}$-errors with $\beta=4$.

Figure 5.6. $L^{2}_{w^{-1}}$-errors with $\beta=4$.

Figure 5.7. $L^{2}$-errors with $\beta=4$.

Figure 5.8. $H^{1}$-errors with $\beta=4$.

Figure 5.9. $L^{2}_{w^{-1}}$-errors with $\beta=4$.

Figure 5.10. $L^{2}$-errors.

###  5.2. Robin boundary value problems

We take $\mu=1$, $\gamma=1$ and $\eta=0$ for the Robin boundary value problem
(3.1) and consider two examples with different decay properties for the exact
solutions.

  * •

$u(x)=e^{-x}(\sin(2x)+\cos(2x)),$ which is exponential decay with oscillation.
In Figures 5.10, 5.12 and 5.12, we plot the $\log_{10}$ of the discrete
$L^{2}$-, $H^{1}$\- and $L^{2}_{(1+w)^{-1}}$-errors vs. $N$ with various
values of $\beta$. Clearly, the approximate solutions converge at exponential
rates.

  * •

$u(x)=(\sin(2x)+\cos(2x))(h+x)^{-h},$ which is algebraic decay with
oscillation. In Figures 5.14, 5.14 and 5.15, we plot the $\log_{10}$ of the
discrete $L^{2}$-, $H^{1}$\- and $L^{2}_{(1+w)^{-1}}$-errors vs.
$\log_{10}(\beta N)$ with fixed $\beta=4$ and various values of $h$. Since

$\displaystyle\|(\sin(2x)+\cos(2x))(h+x)^{-h}\|_{r,-1,\beta}<\infty,\quad\forall
0\leq r<2h,$

the expected $L^{2}$\- and $H^{1}$\- (resp. $L^{2}_{(1+w)^{-1}}$-) errors
given by Theorem 4.3 can be bounded by $c(\beta N)^{1/2-h+\varepsilon}$ (resp.
$c(\beta N)^{-h+\varepsilon}$) for any $\varepsilon>0$. Again, the observed
convergence rates plotted in Figures 5.14, 5.14 and 5.15 agree well with the
theoretical results.

Figure 5.11. $H^{1}$-errors.

Figure 5.12. $L^{2}_{(1+w)^{-1}}$-errors.

Figure 5.13. $L^{2}$-errors with $\beta=4$.

Figure 5.14. $H^{1}$-errors with $\beta=4$.

Figure 5.15. $L^{2}_{(1+w)^{-1}}$-errors with $\beta=4$.

###  5.3. On the condition numbers

To demonstrate the essential superiority of our diagonalized Laguerre spectral
method to the classic Laguerre methods, we finally examine the issue on
condition numbers for the resulting algebraic systems. The diagonalized
Laguerre spectral method use the Sobolev-orthogonal Laguerre functions
$\\{\mathcal{S}_{k}^{\beta}(x)/\sqrt{\varrho_{k}}\\}_{k=1}^{N}$ and
$\\{\mathcal{R}_{k}^{\beta}(x)/\sqrt{\rho_{k}}\\}_{k=0}^{N}$ as the basis
functions for (3.8) and (3.3), respectively. All the condition numbers of the
corresponding total stiff matrices are equal to $1$. While in the classical
Laguerre spectral method, the basis functions for (3.8) and (3.3) are chosen
as $\\{xl_{k}^{0,\beta}(x)\\}_{k=0}^{N-1}$ and
$\\{l_{k}^{0,\beta}(x)\\}_{k=0}^{N}$, respectively. The corresponding total
stiff matrices have off-diagonal entries.

In Table 5.1 below, we list the condition numbers of the total stiff matrices
of the classical Laguerre spectral method for (3.6) with $\gamma=1$ versus
various $N$ and $\beta$. The condition numbers of the classical Laguerre
spectral method for (3.1) with $\gamma=1$ and $\mu=1$ are tabulated in Table
5.2. We note that the condition numbers of the resulting systems increase
asymptotically as $\mathcal{O}(N^{2}).$

N | $\beta=1$ | $\beta=2$ | $\beta=3$
---|---|---|---
10 | 1.5899E+03 | 5.1603E+02 | 2.8771E+02
30 | 2.3557E+04 | 6.8162E+03 | 3.5015E+03
50 | 7.5948E+04 | 2.1307E+04 | 1.0654E+04
70 | 1.6112E+05 | 4.4494E+04 | 2.1921E+04
90 | 2.8034E+05 | 7.6647E+04 | 3.7394E+04
110 | 4.3446E+05 | 1.1795E+05 | 5.7137E+04
130 | 6.2409E+05 | 1.6853E+05 | 8.1197E+04
150 | 8.4971E+05 | 2.2850E+05 | 1.0961E+05
Table 5.1. Condition numbers of the classic Laguerre spectral method for (3.6) with $\gamma=1$. N | $\beta=1$ | $\beta=2$ | $\beta=3$
---|---|---|---
10 | 5.8863E+01 | 2.1075E+02 | 4.4972E+02
30 | 4.1557E+02 | 1.6049E+03 | 3.5610E+03
50 | 1.0965E+03 | 4.2960E+03 | 9.5913E+03
70 | 2.1016E+03 | 8.2840E+03 | 1.8540E+04
90 | 3.4309E+03 | 1.3569E+04 | 3.0407E+04
110 | 5.0845E+03 | 2.0151E+04 | 4.5191E+04
130 | 7.0623E+03 | 2.8029E+04 | 6.2894E+04
150 | 9.3643E+03 | 3.7205E+04 | 8.3515E+04
Table 5.2. Condition numbers of the classic Laguerre spectral method for (3.1)
with $\gamma=1$ and $\mu=1$.

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# Norm inequalities related to Heinz and Heron operator means

A. G. Ghazanfari  Department of Mathematics, Lorestan University, P.O. Box
465, Khoramabad, Iran<EMAIL_ADDRESS>
###### Abstract.

In this article we present some new comparisons between the Heinz and Heron
operator means, which improve some recent results known from the literature.
We derive some refinements of these inequalities for unitarily invariant norms
with the help of the contractive maps.

###### Key words and phrases:

Norm inequality, Operator inequality, Heinz mean.

###### 2010 Mathematics Subject Classification:

47A30, 47A63, 15A45.

Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

##  1\. Introduction and preliminaries

Let $M_{m,n}(\mathbb{C})$ be the space of $m\times n$ complex matrices and
$M_{n}(\mathbb{C})=M_{n,n}(\mathbb{C})$. Let $|||.|||$ denote any unitarily
invariant norm on $M_{n}(\mathbb{C})$. So, $|||UAV|||=|||A|||$ for all $A\in
M_{n}(\mathbb{C})$ and for all unitary matrices $U,V\in M_{n}(\mathbb{C})$.
The Hilbert-Schmidt and trace class norm of $A=[a_{ij}]\in M_{n}(\mathbb{C})$
are denoted by

$\|A\|_{2}=\left(\sum_{j=1}^{n}s_{j}^{2}(A)\right)^{\frac{1}{2}},\left\|A\right\|_{1}=\sum_{j=1}^{n}s_{j}(A)$

where $s_{1}(A)\geq s_{2}(A)\geq...\geq s_{n}(A)$ are the singular values of
$A$, which are the eigenvalues of the positive semidefinite matrix $\mid
A\mid=(A^{*}A)^{\frac{1}{2}}$, arranged in decreasing order and repeated
according to multiplicity. For Hermitian matrices $A,B\in M_{n}(\mathbb{C})$,
we write that $A\geqslant 0$ if $A$ is positive semidefinite, $A>0$ if $A$ is
positive definite, and $A\geqslant B$ if $A-B\geqslant 0$.

Let $A,B,X\in M_{n}$, $A,B$ are positive definite, The matrix version of the
Heinz means are defined by

$H_{\nu}(A,X,B)=\frac{A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}}{2},$

and the matrix version of the Heron means are defined by

$\mathcal{F}_{\alpha}(A,X,B)=(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}}+\alpha\left(\frac{AX+XB}{2}\right).$

In 1998, Zhan [11] proved that if $\frac{1}{4}\leq\nu\leq\frac{3}{4}$ and
$\alpha\in[\frac{1}{2},\infty)$, then

$\displaystyle|||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq\frac{2}{2+t}|||AX+XB+tA^{\frac{1}{2}}XB^{\frac{1}{2}}|||,$
(1.1)

for $-2<t\leq 2$.

Kaur and Singh [4] have shown that if $\frac{1}{4}\leq\nu\leq\frac{3}{4}$ and
$\alpha\in[\frac{1}{2},\infty)$, then

$|||H_{\nu}(A,X,B)|||\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(1.2)

Obviously, if $A,B,X\in M_{n}$, such that $A,B$ are positive definite, then
for $\frac{1}{4}\leq\nu\leq\frac{3}{4}$ and $\alpha\in[\frac{1}{2},\infty)$,
and any unitarily invariant norm $|||.|||$, the following inequalities hold

$|||A^{\frac{1}{2}}XB^{\frac{1}{2}}|||\leq|||H_{\nu}(A,X,B)|||\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(1.3)

I. Ali, H. Yang and A. shakoor [1] gave a refinement of the inequality (1.3)
as follows:

$\displaystyle|||H_{\nu}(A,X,B)|||\leq(4r_{0}-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}|||+2(1-2r_{0})\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$

where $\frac{1}{4}\leq\nu\leq\frac{3}{4}$, $\alpha\in[\frac{1}{2},\infty)$ and
$r_{0}=\min\\{\nu,1-\nu\\}$.

Heretofore the inequalities discussed above are proved in the setting of
matrices. Kapil and Singh in [3], using the contractive maps proved that the
relation (1.2) holds for positive definite operators on a complex Hilbert
space $H$. They also proved that if $A,B$ and $X$ are bounded linear operators
on $H$ and $A,B$ are positive definite operators and
$\alpha\in[\frac{1}{2},\infty)$, then

$\left|\left|\left|A^{\frac{1}{2}}XB^{\frac{1}{2}}\right|\right|\right|\leq\left|\left|\left|\int_{0}^{1}A^{\nu}XB^{1-\nu}d\nu\right|\right|\right|\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|,$
(1.4)

the function $|||\mathcal{F}_{\alpha}|||$ is increasing for
$\alpha\in[\frac{1}{2},\infty)$ and
$|||\mathcal{F}_{\alpha}|||\leq|||\mathcal{F}_{\frac{1}{2}}|||$ for
$\alpha\in[0,\frac{1}{2}]$.

Heinz and Heron inequalities involving unitarily invariant norms have received
renewed attention in recent years and a remarkable variety of refinements and
generalizations have been found (see, for example, [2, 5, 6, 7]).

The aim of this paper is to obtain refinements or generalizations of the above
mentioned inequalities in the setting of operators for unitarily invariant
norms.

##  2\.  Main results

Let $B(H)$ denote the set of all bounded linear operators on a complex Hilbert
space $H$. An operator $A\in B(H)$ is positive, and we write $A\geq 0$, if
$(Ax,x)\geq 0$ for every vector $x\in H$. If $A$ and $B$ are self-adjoint
operators, the order relation $A\geq B$ means, as usual, that $A-B$ is a
positive operator.

To reach inequalities for bounded self-adjoint operators on Hilbert space, we
shall use the following monotonicity property for operator functions:
If $X\in B(H)$ is self adjoint with a spectrum $Sp(X)$, and $f,g$ are
continuous real valued functions on an interval containing $Sp(X)$, then

$f(t)\geq g(t),~{}t\in Sp(X)\Rightarrow~{}f(X)\geq g(X).$ (2.1)

For more details about this property, the reader is referred to [9].

Let $L_{X},R_{Y}$ denote the left and right multiplication maps on $B(H)$,
respectively, that is, $L_{X}(T)=XT$ and $R_{Y}(T)=TY$. Since $L_{X}$ and
$R_{Y}$ commute, we have

$e^{L_{X}+R_{Y}}(T)=e^{X}Te^{Y}.$

Let $U$ be an invertible positive operator in $B(H)$, then there exists a
self-adjoint operator $V\in B(H)$ such that $U=e^{V}$. Let $n\in\mathbb{N}$
and $A,B$ be two invertible positive operators in $B(H)$. To simplify
computations, we denote $A$ and $B$ by $e^{2X_{1}}$ and $e^{2Y_{1}}$,
respectively, where $X_{1}$ and $Y_{1}$ in $B(H)$ are self-adjoint. The
corresponding operator map $L_{X_{1}}-R_{Y_{1}}$ is denoted by $D$. With these
notations, we now use the results proved in [3, 8] to derive the Heinz and
Heron type inequalities for unitarily invariant norms.

######  Proposition 1.

Let $0\leq s_{2}\leq s_{1}$ and $r,r^{\prime}\leq\frac{s_{1}+s_{2}}{2}$ or
$s_{1}\leq s_{2}\leq 0$ and $r,r^{\prime}\geq\frac{s_{1}+s_{2}}{2}$, then the
following operator maps are contractive,

  1. (1)

$\frac{(1+t)\cosh(rD)}{\cosh(s_{1}D)+t\cosh(s_{2}D)}$, for $-1<t\leq 1$.

  2. (2)

$\frac{\alpha\cosh(rD)+(1-\alpha)\cosh(r^{\prime}D)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}$,
for $0\leq\alpha\leq 1,~{}\frac{1}{2}\leq\beta$.

  3. (3)

$\frac{(1+t)\sinh(rD)}{r(\sinh(s_{1}D)+t\sinh(s_{2}D))}$, for $-1<t\leq 1$ and
$|s_{1}+s_{2}|\geq 2$.

  4. (4)

$\frac{\frac{\alpha}{r}\sinh(rD)+\frac{1-\alpha}{r^{\prime}}\sinh(r^{\prime}D)}{\beta\sinh(s_{1}D)+(1-\beta)\sinh(s_{2}D)}$,
for $0\leq\alpha\leq 1,~{}\frac{1}{2}\leq\beta$ and $|s_{1}+s_{2}|\geq 2$.

###### Proof.

  1. (1)

It is easy to see that

$\displaystyle\frac{(1+t)\cosh(rD)}{\cosh(s_{1}D)+t\cosh(s_{2}D)}$
$\displaystyle=\frac{(1+t)\cosh(rD)}{\cosh(s_{1}D)+\cosh(s_{2}D)}\left(1-\frac{(1-t)\cosh(s_{2}D)}{\cosh(s_{1}D)+\cosh(s_{2}D)}\right)^{-1}$
$\displaystyle=\frac{1+t}{2}\frac{2\cosh(rD)}{\cosh(s_{1}D)+\cosh(s_{2}D)}\times\sum_{n=0}^{\infty}\left(\frac{1-t}{2}\right)^{n}\left(\frac{2\cosh(s_{2}D)}{\cosh(s_{1}D)+\cosh(s_{2}D)}\right)^{n}.$

For each $T\in B(H)$, using part (2) of Corollary 2.5 of [3], we obtain

$\displaystyle\left|\left|\left|\frac{(1+t)\cosh(rD)}{\cosh(s_{1}D)+t\cosh(s_{2}D)}T\right|\right|\right|\leq\frac{1+t}{2}\sum_{n=0}^{\infty}\left(\frac{1-t}{2}\right)^{n}|||T|||=|||T|||.$

  2. (2)

To prove part (2), we write

$\displaystyle\frac{\alpha\cosh(rD)+(1-\alpha)\cosh(r^{\prime}D)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}$
$\displaystyle=\alpha\frac{\cosh(rD)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}+(1-\alpha)\frac{\cosh(r^{\prime}D)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}$

Now, using part (1), we obtain

$\left|\left|\left|\frac{\cosh(rD)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}T\right|\right|\right|\\\
=\left|\left|\left|\frac{\frac{1}{\beta}\cosh(rD)}{\cosh(s_{1}D)+(\frac{1}{\beta}-1)\cosh(s_{2}D)}T\right|\right|\right|\leq
1.$

Similarly,

$\left|\left|\left|\frac{\cosh(r^{\prime}D)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}T\right|\right|\right|\leq
1.$

Consequently,

$\displaystyle\left|\left|\left|\frac{\alpha\cosh(rD)+(1-\alpha)\cosh(r^{\prime}D)}{\beta\cosh(s_{1}D)+(1-\beta)\cosh(s_{2}D)}T\right|\right|\right|\leq
1.$

A simple calculation shows that

$\displaystyle\frac{(1+t)\sinh(rD)}{r(\sinh(s_{1}D)+t\sinh(s_{2}D))}$
$\displaystyle=\frac{1+t}{s_{1}+s_{2}}\frac{(s_{1}+s_{2})\sinh(rD)}{r(\sinh(s_{1}D)+\sinh(s_{2}D))}\left(1-\frac{(1-t)\sinh(s_{2}D)}{\sinh(s_{1}D)+\sinh(s_{2}D)}\right)^{-1}$
$\displaystyle=\frac{1+t}{s_{1}+s_{2}}\frac{(s_{1}+s_{2})\sinh(rD)}{r(\sinh(s_{1}D)+\sinh(s_{2}D))}$
$\displaystyle\times\sum_{n=0}^{\infty}\left(\frac{2s_{2}}{s_{1}+s_{2}}\right)^{n}\left(\frac{1-t}{2}\right)^{n}\left(\frac{(s_{1}+s_{2})\sinh(s_{2}D)}{s_{2}(\sinh(s_{1}D)+\sinh(s_{2}D))}\right)^{n}.$

Using part (3) of Corollary 2.5 in [3], and $|\frac{2s_{2}}{s_{1}+s_{2}}|\leq
1$, we deduce for each $T\in B(H)$

$\displaystyle\left|\left|\left|\frac{(1+t)\sinh(rD)}{r(\sinh(s_{1}D)+t\sinh(s_{2}D))}T\right|\right|\right|$
$\displaystyle\leq\frac{1+t}{|s_{1}+s_{2}|}\sum_{n=0}^{\infty}\left(\frac{1-t}{2}\right)^{n}|||T|||$
$\displaystyle=\frac{2}{|s_{1}+s_{2}|}|||T|||\leq|||T|||.$

The proof of (4) is similar to (2) and we omitted it. ∎

######  Theorem 1.

Let $A,X,B\in B(H)$ and $A,B$ be two positive definite operators. Let
$\frac{1}{2}\leq\beta\leq 1$ and $\alpha\geq\frac{1}{2}$.

  1. (1)

If $\frac{3}{8}\leq\nu\leq\frac{5}{8}$, then

$\displaystyle|||H_{\nu}(A,X,B)|||$
$\displaystyle\leq\left|\left|\left|(1-\beta)A^{\frac{1}{2}}XB^{\frac{1}{2}}+\beta
H_{\frac{1}{4}}(A,X,B)\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(2.2)

  2. (2)

If $\frac{5}{16}\leq\nu\leq\frac{11}{16}$, then

$\displaystyle|||H_{\nu}(A,X,B)|||$
$\displaystyle\leq\left|\left|\left|(1-\beta)H_{\frac{3}{8}}(A,X,B)+\beta
H_{\frac{1}{4}}(A,X,B)\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(2.3)

###### Proof.

Put $T=A^{\frac{1}{2}}XB^{\frac{1}{2}}$, a calculus computation shows that

$\displaystyle
A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}=A^{\nu-\frac{1}{2}}A^{\frac{1}{2}}XB^{\frac{1}{2}}B^{\frac{1}{2}-\nu}+A^{\frac{1}{2}-\nu}A^{\frac{1}{2}}XB^{\frac{1}{2}}B^{\nu-\frac{1}{2}}$
$\displaystyle=A^{\nu-\frac{1}{2}}TB^{\frac{1}{2}-\nu}+A^{\frac{1}{2}-\nu}TB^{\nu-\frac{1}{2}}$
$\displaystyle=e^{(2\nu-1)X_{1}}Te^{(1-2\nu)Y_{1}}+e^{(1-2\nu)X_{1}}Te^{(2\nu-1)Y_{1}}$
$\displaystyle=e^{(2\nu-1)D}T+Te^{-(2\nu-1)D}=2\cosh((2\nu-1)D)T.$

Hence, we can write the operator on the left hand side of the first inequality
in ((1)) as

$\displaystyle\frac{A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}}{2}=\cosh((2\nu-1)D)T$
(2.4)

and the operator on the right hand side of the first inequality in ((1)) as

$\displaystyle(1-\beta)A^{\frac{1}{2}}XB^{\frac{1}{2}}+\beta\frac{A^{\frac{1}{4}}XB^{\frac{3}{4}}+A^{\frac{3}{4}}XB^{\frac{1}{4}}}{2}=\left((1-\beta)+\beta\cosh\left(\frac{1}{2}D\right)\right)T.$

Considering

$\displaystyle\frac{\cosh((2\nu-1)D)}{(1-\beta)+\beta\cosh(\frac{1}{2}D)}=\frac{\frac{1}{\beta}\cosh((2\nu-1)D)}{\frac{(1-\beta)}{\beta}+\cosh(\frac{1}{2}D)}$

we put $t=\frac{1}{\beta}-1$, then $|t|\leq
1\Leftrightarrow\beta\geq\frac{1}{2}$. Therefore, part (1) of Proposition 1
proves the first inequality in ((1)).

$\left|\left|\left|(1-\beta)A^{\frac{1}{2}}XB^{\frac{1}{2}}+\beta
H_{\frac{1}{4}}(A,X,B)\right|\right|\right|\\\
\leq(1-\beta)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}|||+\beta|||H_{\frac{1}{4}}(A,X,B)|||\\\
\leq(1-\beta)|||\mathcal{F}_{\alpha}(A,X,B)|||+\beta|||\mathcal{F}_{\alpha}(A,X,B)|||.$
(2.5)

Hence the proof of part (1) is completed.

Using the same arguments, we can show the operator on the right hand side of
the first inequality in ((2)) as

$\displaystyle(1-\beta)\frac{A^{\frac{3}{8}}XB^{\frac{5}{8}}+A^{\frac{5}{8}}XB^{\frac{3}{8}}}{2}+\beta\frac{A^{\frac{1}{4}}XB^{\frac{3}{4}}+A^{\frac{3}{4}}XB^{\frac{1}{4}}}{2}$
$\displaystyle=\left((1-\beta)\cosh\left(\frac{1}{4}D\right)+\beta\cosh\left(\frac{1}{2}D\right)\right)T.$
(2.6)

By part (1) of Proposition 1 the function
$\frac{\cosh((2\nu-1)D)}{(1-\beta)\cosh\left(\frac{1}{4}D\right)+\beta\cosh\left(\frac{1}{2}D\right)}$
is contractive, therefore the first inequality in ((2)) holds. The Similar to
relation (2.5), the second inequality in ((2)) is proved. Hence the proof is
completed.

∎

######  Theorem 2.

Let $A,X,B\in B(H)$ and $A,B$ be two positive definite operators. Let
$\frac{1}{2}\leq\beta,\gamma,\delta\leq 1$ and $\alpha\geq\frac{1}{2}$.

  1. (1)

If $\frac{9}{32}\leq\nu\leq\frac{23}{32}$, then

$\displaystyle|||H_{\nu}(A,X,B)|||$
$\displaystyle\leq\left|\left|\left|(1-\beta)H_{\frac{5}{16}}(A,X,B)+\beta
H_{\frac{1}{4}}(A,X,B)\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(2.7)

  2. (2)

If $\frac{11}{32}\leq\nu\leq\frac{21}{32}$, then

$\displaystyle|||H_{\nu}(A,X,B)|||$
$\displaystyle\leq\left|\left|\left|(1-\gamma)H_{\frac{3}{8}}(A,X,B)+\gamma
H_{\frac{5}{16}}(A,X,B)\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|(1-\beta)H_{\frac{3}{8}}(A,X,B)+\beta
H_{\frac{1}{4}}(A,X,B)\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(2.8)

###### Proof.

By the same arguments and notations used earlier, we can show the operator on
the right hand side of the first inequality in ((1)) as

$\displaystyle(1-\beta)\frac{A^{\frac{5}{16}}XB^{\frac{11}{16}}+A^{\frac{11}{16}}XB^{\frac{5}{16}}}{2}+\beta\frac{A^{\frac{1}{4}}XB^{\frac{3}{4}}+A^{\frac{3}{4}}XB^{\frac{1}{4}}}{2}$
$\displaystyle=\left((1-\beta)\cosh\left(\frac{3}{8}D\right)+\beta\cosh\left(\frac{1}{2}D\right)\right)T.$

Being contractive the map
$\frac{\cosh((2\nu-1)D)}{(1-\beta)\cosh\left(\frac{3}{8}D\right)+\beta\cosh\left(\frac{1}{2}D\right)}$
proves the first inequality in ((1)). The Similar to relation (2.5), the
second inequality in ((1)) is proved.

It is easy to see that

$\displaystyle(1-\gamma)\frac{A^{\frac{3}{8}}XB^{\frac{5}{8}}+A^{\frac{5}{8}}XB^{\frac{3}{8}}}{2}+\gamma\frac{A^{\frac{5}{16}}XB^{\frac{11}{16}}+A^{\frac{11}{16}}XB^{\frac{5}{16}}}{2}$
$\displaystyle=\left((1-\gamma)\cosh\left(\frac{1}{4}D\right)+\gamma\cosh\left(\frac{3}{8}D\right)\right)T.$

The maps
$\frac{\cosh((2\nu-1)D)}{(1-\gamma)\cosh\left(\frac{1}{4}D\right)+\gamma\cosh\left(\frac{3}{8}D\right)}$
and
$\frac{(1-\gamma)\cosh\left(\frac{1}{4}D\right)\gamma\cosh\left(\frac{3}{8}D\right)}{(1-\beta)\cosh\left(\frac{1}{4}D\right)\beta\cosh\left(\frac{1}{2}D\right)}$
are contractive. Therefore we get the desired results in part (2). ∎

In the following Theorem, we give a refinement of (1.4).

######  Theorem 3.

Let $A,X,B\in B(H)$ and $A,B$ be two positive definite operators. If
$\frac{1}{4}<\nu<\frac{3}{4}$ and $\alpha\geq\frac{1}{2}$, then

$\displaystyle|||H_{\nu}(A,X,B)|||$
$\displaystyle\leq\left|\left|\left|\frac{1}{2}H_{\frac{1}{8}}(A,X,B)+\frac{1}{2}H_{\frac{3}{8}}(A,X,B)\right|\right|\right|\leq\left|\left|\left|\int_{0}^{1}A^{t}XB^{1-t}dt\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|\frac{1}{2}H_{\frac{1}{4}}(A,X,B)+\frac{1}{2}F_{\frac{1}{2}}(A,X,B)\right|\right|\right|$
$\displaystyle\leq\left|\left|\left|\mathcal{F}_{\alpha}(A,X,B)\right|\right|\right|.$
(2.9)

###### Proof.

By the relation ((2)), we have

$\displaystyle H_{\nu}(A,X,B)=\cosh((2\nu-1)D)T.$

Again as in previous theorems, we get

$\displaystyle\frac{1}{2}H_{\frac{1}{8}}(A,X,B)+\frac{1}{2}H_{\frac{3}{8}}(A,X,B)$
$\displaystyle=\left(\frac{1}{2}\cosh\left(\frac{3}{4}D\right)+\frac{1}{2}\cosh\left(\frac{1}{4}D\right)\right)T,$

$\displaystyle\int_{0}^{1}A^{\nu}XB^{1-\nu}d\nu=\int_{0}^{1}\exp((2\nu-1)D)Td\nu=(D^{-1}\sinh
D)T,$ (2.10)

and

$\displaystyle\frac{1}{2}H_{\frac{1}{4}}(A,X,B)+\frac{1}{2}F_{\frac{1}{2}}(A,X,B)=\left(\frac{1}{2}\cosh\left(\frac{1}{2}D\right)+\frac{1}{2}\left(\frac{1+\cosh(D)}{2}\right)\right)T.$

We get the first inequality in (3), since the map
$\frac{2\cosh(2\nu-1)D}{\cosh(\frac{3}{4}D)+\cosh(\frac{1}{4}D)}$ is
contractive.

By part (3) of the Proposition 2.3 in [3], the map
$\frac{D(\cosh(\frac{3}{4}D)+\cosh(\frac{1}{4}D))}{2\sinh
D}=\frac{D\cosh(\frac{1}{4}D)}{2\sinh D}$ is contractive. This proves the
second inequality in (3).

By part (4) of the Proposition 2.3 in [3], the following map is contractive.

$\displaystyle\frac{2\sinh
D}{D\left(\cosh\left(\frac{1}{2}D\right)+\frac{1+\cosh(D)}{2}\right)}=\frac{2\sinh
D}{D\left(\cosh\left(\frac{1}{2}D\right)+\cosh^{2}(\frac{1}{2}D)\right)}$
$\displaystyle=\frac{2\sinh(\frac{1}{2}D)\cosh(\frac{1}{2}D)}{D\left(\cosh\left(\frac{1}{2}D\right)\cosh^{2}(\frac{1}{4}D)\right)}=\frac{4\sinh(\frac{1}{4}D)}{D\cosh\left(\frac{1}{4}D\right)}.$

This proves the third inequality in (3).

$\displaystyle\left|\left|\left|\frac{1}{2}H_{\frac{1}{4}}(A,X,B)+\frac{1}{2}F_{\frac{1}{2}}(A,X,B)\right|\right|\right|\leq\frac{1}{2}|||H_{\frac{1}{4}}(A,X,B)|||+\frac{1}{2}|||F_{\frac{1}{2}}(A,X,B)|||$
$\displaystyle\leq\frac{1}{2}|||F_{\alpha}(A,X,B)|||+\frac{1}{2}|||F_{\alpha}(A,X,B)|||=|||F_{\alpha}(A,X,B)|||.$

This completes the proof. ∎

As an application, we give the following inequalities for unitarily invariant
norms. From inequality (1.2), we get

$\displaystyle\left|\left|\left|\int_{\frac{1}{4}}^{\frac{3}{4}}H_{\nu}(A,X,B)d\nu\right|\right|\right|$
$\displaystyle\leq\int_{\frac{1}{4}}^{\frac{3}{4}}|||H_{\nu}(A,X,B)|||d\nu$
$\displaystyle\leq\int_{\frac{1}{4}}^{\frac{3}{4}}|||F_{\alpha}(A,X,B)|||d\nu=\frac{1}{2}|||F_{\alpha}(A,X,B)|||.$

Hence from inequalities ((1)), ((1)) and ((2)) respectively, we get

$\displaystyle\left|\left|\left|\int_{\frac{3}{8}}^{\frac{5}{8}}H_{\nu}(A,X,B)d\nu\right|\right|\right|\leq\frac{1}{4}|||F_{\alpha}(A,X,B)|||,$
$\displaystyle\left|\left|\left|\int_{\frac{9}{32}}^{\frac{23}{32}}H_{\nu}(A,X,B)d\nu\right|\right|\right|\leq\frac{7}{16}|||F_{\alpha}(A,X,B)|||$

and

$\displaystyle\left|\left|\left|\int_{\frac{11}{32}}^{\frac{21}{32}}H_{\nu}(A,X,B)d\nu\right|\right|\right|\leq\frac{5}{16}|||F_{\alpha}(A,X,B)|||.$

As other applications, we now consider some inequalities that involve
differences and sums of operators. These inequalities in the operator setting
can be seen in [3], and in the matrix setting can be seen in [10]. In the
following theorems, we give new forms of the presented inequalities in Theorem
4.4 and Theorem 4.10 in [10], and inequality (3.25) in [3]. We also give
complementary inequalities to them.

Let $p\in\mathbb{R}$, $A,X,B\in B(H)$ and $A,B$ be two positive definite
operators. For $p\in\mathbb{R}$, we introduce the function
$f(\nu)=|||A^{\nu}XB^{p-\nu}+A^{p-\nu}XB^{\nu}|||$. It is easy to show that
the function $f(\nu)$ is decreasing for $\nu\leq\frac{p}{2}$, and increasing
for $\nu\geq\frac{p}{2}$. The part (2) of Proposition 1, implies that $f(\nu)$
is convex on $\mathbb{R}$.

######  Theorem 4.

Let $r,p,\nu\in\mathbb{R}$, $~{}A,X,B\in B(H)$ and $A,B$ be two positive
definite operators. If $\nu\leq 2r\leq p$ or $\nu\geq 2r\geq p$, then for
$|t|\leq 1$

$\displaystyle(1)~{}(1+t)$ $\displaystyle|||A^{r}XB^{p-r}+A^{p-r}XB^{r}|||$
$\displaystyle\leq|||A^{p}X+t(A^{\nu}XB^{p-\nu}+A^{p-\nu}XB^{\nu})+XB^{p})|||.$
(2.11)

Furthermore, if $|p-\nu|\geq 1$, then

$\displaystyle(2$ $\displaystyle)~{}(1+t)|||A^{r}XB^{p-r}-A^{p-r}XB^{r}|||$
$\displaystyle\leq|p-2r|~{}|||A^{p}X+t(A^{\nu}XB^{p-\nu}-A^{p-\nu}XB^{\nu})-XB^{p})|||.$
(2.12)

###### Proof.

  1. (1)

Notice that

$\displaystyle
A^{\nu}XB^{p-\nu}+A^{p-\nu}XB^{\nu}=A^{\nu-\frac{p}{2}}A^{\frac{p}{2}}XB^{\frac{p}{2}}B^{\frac{p}{2}-\nu}+A^{\frac{p}{2}-\nu}A^{\frac{p}{2}}XB^{\frac{p}{2}}B^{\nu-\frac{p}{2}}.$

Put $U=A^{\frac{p}{2}}XB^{\frac{p}{2}}$. Then

$\displaystyle A^{\nu}XB^{p-\nu}+A^{p-\nu}XB^{\nu}$
$\displaystyle=A^{\nu-\frac{p}{2}}UB^{\frac{p}{2}-\nu}+A^{\frac{p}{2}-\nu}UB^{\nu-\frac{p}{2}}$
$\displaystyle=2(\cosh(p-2\nu)D)U,$

$\displaystyle
A^{p}X+XB^{p}=A^{\frac{p}{2}}UB^{-\frac{p}{2}}+A^{-\frac{p}{2}}UB^{\frac{p}{2}}=2(\cosh
pD)U.$

Using part (1) of Proposition 1, we get the following map is contractive.

$\displaystyle\frac{(1+t)\cosh((p-2r)D)}{\cosh(pD)+t\cosh(p-2\nu)D}.$

  2. (2)

We know that

$\displaystyle
A^{\nu}XB^{p-\nu}-A^{p-\nu}XB^{\nu}=A^{\nu-\frac{p}{2}}A^{\frac{p}{2}}XB^{\frac{p}{2}}B^{\frac{p}{2}-\nu}-A^{\frac{p}{2}-\nu}A^{\frac{p}{2}}XB^{\frac{p}{2}}B^{\nu-\frac{p}{2}}.$

Put $U=A^{\frac{p}{2}}XB^{\frac{p}{2}}$. Then

$\displaystyle A^{\nu}XB^{p-\nu}-A^{p-\nu}XB^{\nu}$
$\displaystyle=A^{\nu-\frac{p}{2}}UB^{\frac{p}{2}-\nu}-A^{\frac{p}{2}-\nu}UB^{\nu-\frac{p}{2}}$
$\displaystyle=2(\sinh(p-2\nu)D)U,$

$\displaystyle
A^{p}X-XB^{p}=A^{\frac{p}{2}}UB^{-\frac{p}{2}}-A^{-\frac{p}{2}}UB^{\frac{p}{2}}=2(\sinh
pD)U.$

Using part (3) of Proposition 1, we get the following map is contractive.

$\displaystyle\frac{(1+t)\sinh((p-2r)D)}{(p-2r)(\sinh(pD)+t\sinh(p-2\nu)D)}.$

∎

In special case, the relation (4) for $p=1$ and $\nu=\frac{1}{2}$, becomes the
following inequality

$\displaystyle|||A^{r}X$ $\displaystyle
B^{1-r}+A^{1-r}XB^{r}|||\leq\frac{2}{2+t}\left|\left|\left|tA^{\frac{1}{2}}XB^{\frac{1}{2}}+\frac{AX+XB}{2}\right|\right|\right|,$

which holds for $\frac{1}{4}\leq r\leq\frac{3}{4}$ and $-2<t\leq 2$. In this
case the relation (1.1) follows from (4).

######  Theorem 5.

Let $r,p,\nu\in\mathbb{R}$, $~{}A,X,B\in B(H)$ and $A,B$ be two positive
definite operators. If $r\leq 0\leq p$ and $r\leq\nu\leq\frac{p}{2}$ (or,
$r\geq 0\geq p$ and $r\geq\nu\geq\frac{p}{2}$), then for $t\geq 0$,

$\displaystyle(1)~{}(1+t)~{}|||A^{r}X$ $\displaystyle
B^{p-r}+A^{p-r}XB^{r}|||$
$\displaystyle\geq|||A^{p}X+t(A^{\nu}XB^{p-\nu}+A^{p-\nu}XB^{\nu})+XB^{p})|||,$
(2.13)

and

$\displaystyle(2)~{}\left(\frac{(1+t)p-2t\nu}{p-2r}\right)$
$\displaystyle|||A^{r}XB^{p-r}-A^{p-r}XB^{r}|||$
$\displaystyle\geq|||A^{p}X+t(A^{\nu}XB^{p-\nu}-A^{p-\nu}XB^{\nu})-XB^{p})|||.$
(2.14)

###### Proof.

First, we prove inequality (5). By part (4) of the Proposition 2.4 in [3], we
get

$\displaystyle|||A^{p}X-XB^{p}|||\leq\left(\frac{p}{p-2r}\right)|||A^{r}XB^{p-r}-A^{p-r}XB^{r}|||,$

and

$\displaystyle|||A^{\nu}XB^{p-\nu}-A^{p-\nu}XB^{\nu}|||\leq\left(\frac{p-2\nu}{p-2r}\right)|||A^{r}XB^{p-r}-A^{p-r}XB^{r}|||.$

From the following triangle inequality

$\displaystyle|||A^{p}X+t(A^{\nu}XB^{p-\nu}$
$\displaystyle-A^{p-\nu}XB^{\nu})-XB^{p})|||$
$\displaystyle\leq|||A^{p}X-XB^{p}|||+t~{}|||A^{\nu}XB^{p-\nu}-A^{p-\nu}XB^{\nu}|||,$

we deduce the inequality (5). Similarly, the inequality (5) is proved. ∎

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