# Interaction-Range Effects and Universality in the BCS-BEC Crossover of Spin-
Orbit Coupled Fermi Gases

D. Giambastiani  M. Barsanti  M. L. Chiofalo

###### Abstract

We explore the evolution of a ultracold quantum gas of interacting fermions
crossing from a Bardeen-Cooper-Schrieffer (BCS) superfluidity to a Bose-
Einstein condensation (BEC) of molecular bosons in the presence of a tunable-
range interaction among the fermions and of an artificial magnetic field,
which can be used to simulate a pseudo-spin-orbit coupling (SOC) and to
produce topological states. We find that the crossover is affected by a
competition between the finite range of the interaction and the SOC and that
the threshold $\lambda_{B}$ for the topological transition is affected by the
interactions only in the small pair size, BEC-like, regime. Below
$\lambda_{B}$, we find persistence of universal behavior in the critical
temperature, chemical potential, and condensate fraction, provided that the
pair correlation length is used as a driving parameter. Above threshold,
universality is lost in the regime of large pair sizes. Here, the limiting
ground state departs from a weakly-interacting BCS-like, so that a different
description is required. Our results can be relevant in view of current
experiments with cold atoms in optical cavities, where tunable-range effective
atomic interactions can be engineered.

##  1 Introduction

The evolution of superfluidity in a quantum gas of interacting fermionic
particles crossing over from a Bose-Einstein condensation (BEC) of composite
bosons to a Bardeen-Cooper-Schrieffer (BCS) [1] superfluidity of Cooper pairs
[2], is still an open problem. The study of this smooth evolution between a
BCS to BEC state of superfluidity, proposed first by Leggett [3] and further
explored by Nozières and Schmitt-Rink (NSR) [4], has opened the way to a
number of theoretical and numerical efforts aimed to describe the underlying
physics via a universal parameter independent of the details of the
interaction between fermions. After a celebrated log-log plot by Uemura et al.
[5], in which a classification of different conventional and unconventional
superconductors was proposed plotting the respective Fermi temperature vs. the
critical one, the BCS-BEC crossover has become a useful framework to better
understand high-temperature superconductivity [6]. In this context, Pistolesi
and Strinati [7] have introduced the pair-correlation length, representing the
typical size of the fermion pairs, as the universal parameter to describe the
crossover physics. In particular, they noticed that unconventional
superconductors were characterized by shorter correlation lengths, though not
entering the bosonic side of the crossover.

Quantum atomic gases [8, 9, 10, 11] represent an ideal platform where the BCS-
BEC crossover can be investigated, with the advantage of extreme precision and
control typical of atomic physics. In fact, besides the tuning of temperature,
quantum gases offer the possibility of tuning the dimensions from three down
to zero, the interactions in strength and sign via the Fano-Feshbach mechanism
[12] driven by magnetic or optical means, or else in range via the use of
dipolar gases or Rydberg atoms, as well as accessing both internal, spin-like,
and motional degrees of freedom [13]. More recently, in Lev’s group it has
been demonstrated the possibility of tuning the range of effective
interactions among atoms in optical cavities [14].

Current experimental advancements have opened the possibility of studying the
BCS-BEC crossover of quantum gases, in the presence of an external interaction
term formally equivalent to a (so-called artificial) magnetic field [15],
along with the possibility of realizing different kinds of couplings between
atomic internal (e.g. spin or pseudo-spin variables) and external (e.g.
momentum) degrees of freedom, via atom-light interactions induced by laser
lights in bosonic [16, 17] and fermionic [18, 19] systems. Quantum gases with
artificial magnetic fields can be especially interesting to simulate the
Fractional Quantum Hall effect [16] whereas, with spin-orbit coupling, to
simulate topological insulators and superconductors [20], and to boost
applications for measurements of tiny forces, where the coupling between
internal and motional degrees of freedom often represents a crucial tool.
Therefore, quantum gases represent a very convenient experimental and
theoretical laboratory to probe the BCS-BEC crossover idea against different
microscopic models.

Spin-orbit coupling (SOC) physics in the BCS-BEC crossover has been
theoretically investigated for the first time by Vyasanakere and Shenoy et al.
[21] in the presence of a contact interaction. They found that SOC induces the
occurrence of a bound state, in both the BEC and BCS regimes, and leads to a
change in the topology of the Fermi sphere. Among the different SOC types
considered, the symmetric and oblate cases have been found the most
interesting for the crossover physics with respect to the prolate setting.
This topic has been the subject of further investigations of the critical
temperature for superfluidity within the so-called pairing approximation
originally introduced by Kadanoff and Martin [22], in which the non-condensed
fermionic pairs are treated within a number-conserving scheme [23]. The
pairing approximation has been further developed and extensively used by Levin
et al. [24] in connection with high-temperature superconductivity, to explain
the occurrence of a pseudogap in the single-particle excitation spectrum in
the form of a shift of spectral weight towards finite frequencies, and also to
determine the spin and density response behavior of quantum gases with SOC in
the presence of a contact interaction [25].

In this paper, we address two open questions in the physics of interacting
quantum gases with SOC in the symmetric case. First, motivated by current
experiments with cold atoms in optical cavities [14], we investigate the
effects interactions tunable in strength and range. Second, we discuss the
extent to which universality [7] may persist in the crossover once SOC is
introduced. We find that SOC and finite interaction range compete in
determining the crossover physics: the former tends to create more tightly
bound pairs with small size, while longer-range and stronger interactions
favour the formation of larger-sized Cooper pairs. We also study how the
threshold $\lambda_{B}$ of SOC strength [21], above which the Fermi sphere
undergoes a change of topology, is shifted while varying the interactions
strength and range. As to the universal behavior, we find that this persists
for small $\lambda$ values of the SOC coupling, provided that the correlation
length is used as a driving parameter to embody both the strength and range of
the atomic interactions [7]. As expected instead, for larger $\lambda$ values
well beyond the threshold for the topological change of the Fermi sphere, the
fluid crosses over from a BEC-like regime towards a regime characterized by
large-size pairs though significantly different from a weakly interacting BCS
state. The persistence of universal behaviour in the crossover of fermions
with SOC below the threshold for the topological transition, represents a
relevant concept fostering a deeper understanding of the crossover paradigm
and its microscopic implementations, which had not been evidenced so far.

The paper is organized as follows. After introducing the model and having
defined the observables to be investigated, we present our results. We begin
with the effect of the interaction strength and range on the behavior of the
two main parameters of the theory, i.e. the correlation length and the SOC
threshold strength $\lambda_{B}$. Then, the universal behavior of the
observables is discussed. Finally, we summarize our concluding remarks and
future perspectives.

##  2 Model and self-consistent equations

We consider the Hamiltonian for a fluid of interacting fermionic atoms with an
attractive potential $V$ and a symmetric Spin-Orbit-Coupling (S-SOC) with
coupling strength $\lambda$. This can be cast in the compact form [21]:

$\begin{split}\mathcal{H}=&\sum_{\bm{k}}\psi_{\bm{k}}^{\dagger}\left[\left(\frac{\hbar^{2}\bm{k}^{2}}{2m}-\mu\right)\mathbf{1}+\hbar^{2}\lambda
v_{F}\vec{\bm{\sigma}}\bm{k}\right]\psi_{\bm{k}}\\\
-&\frac{1}{\mathcal{V}}\sum_{\bm{k},\bm{k}^{\prime},\bm{q}}V_{\bm{k}\bm{k}^{\prime}}c_{\bm{k}+\frac{\bm{q}}{2}\uparrow}^{\dagger}c_{-\bm{k}+\frac{\bm{q}}{2}\downarrow}^{\dagger}c_{-\bm{k}^{\prime}+\frac{\bm{q}}{2}\downarrow}c_{\bm{k}^{\prime}+\frac{\bm{q}}{2}\uparrow},\end{split}$
(1)

where $\mathcal{V}$ is the volume, $m$ the fermion mass, $\mu$ the chemical
potential, $\vec{\bm{\sigma}}$ is the bmor of Pauli matrices
$\vec{\bm{\sigma}}\equiv(\sigma_{x},\sigma_{y},\sigma_{z})$, $c$ and
$c^{\dagger}$ are the destruction and creation fermionic operators,
respectively, and
$\psi_{\bm{k}}\equiv(c_{\bm{k}\uparrow},c_{\bm{k}\downarrow})^{T}$ the Nambu-
Gor’kov spinor. In our treatment, $V_{\bm{k}\bm{k}^{\prime}}$ is a finite-
range potential, modeled in a separable form as in Nozières and Schmitt-Rink
[4]:

$V_{\bm{k}\bm{k}^{\prime}}=g\frac{4\pi}{mk_{0}k_{F}}w_{\bm{k}}w_{\bm{k}^{\prime}},\qquad
w_{\bm{k}}=\sqrt{\frac{(k_{0}k_{F})^{2}}{(k_{0}k_{F})^{2}+|\bm{k}|^{2}}}.$ (2)

Here and in the following, $k_{F}$, $v_{F}$ and $E_{F}$ are respectively the
Fermi momentum, velocity and energy, and the system adimensional parameters of
the theory are $g>0$ driving the atom interaction strength, $k_{0}^{-1}$
measuring the interaction range, and $\lambda$ the SOC coupling strength.

In fact, the SOC term introduces the special motional-related direction
$\bm{k}$, so that one may distinguish between parallel and antiparallel spins
with respect to $\bm{k}$. The physics can thus be expressed in the so-called
helicity basis:

$\begin{cases}&a_{\bm{k},+}=N_{\perp}^{+}(s_{+}c_{\bm{k}\uparrow}+e^{-i\varphi_{\bm{k}}}c_{\bm{k}\downarrow})\\\
&a_{\bm{k},-}=N_{\perp}^{+}(e^{i\varphi_{\bm{k}}}c_{\bm{k}\uparrow}-s_{+}c_{\bm{k}\downarrow})\end{cases},$
(3)

in terms of the size of the $k$-bmor
$k_{\perp}\equiv\sqrt{k_{x}^{2}+k_{y}^{2}}$ perpendicular to $\hat{z}$, the
relative amplitude $s_{\pm}\equiv(|\bm{k}|\pm k_{z})/k_{\perp}$ of the
parallel ($s_{+}$) and antiparallel ($s_{-}$) components with phase
$\varphi_{\bm{k}}\equiv arg(k_{x}+ik_{y})$, and overall normalization factor
$N_{\perp}^{\pm}\equiv{[2|\bm{k}|(|\bm{k}|\pm
k_{z})]^{-{1}/{2}}}/{k_{\perp}}$. This representation naturally leads to the
description of the available fermionic states, in terms of two Fermi spheres
of different helicity, i.e. different eigenvalues of the operator
$\vec{\bm{\sigma}}\bm{k}$ [21]. In particular, below a critical value
$\lambda_{T}$ of the SOC parameter ($\lambda<\lambda_{T}$) the $-$ helicity
sphere expands with increasing values of $\lambda$, while the $+$ helicity
shrinks until it disappears at $\lambda=\lambda_{T}$. Above the threshold
$\lambda_{T}$, a hole appears in the $-$ helicity Fermi sphere and, as a
result, $-$ helicity fermions fill in the shell between two spheres [21].

In order to describe the physics at finite temperature including the non-
condensed pairs, we resort to the mean-field conserving approximation
introduced by Kadanoff and Martin [22] and subsequently developed by Levin et
al. under the name of pairing approximation [24]. In our case with the
separable potential $V_{\bm{k}\bm{k}^{\prime}}$ and SOC in the helicity basis,
the pairing approximation consists in solving the following self-consistent
set of equations for the T-matrix $t(Q)$ embodying the response $\chi(Q)$, the
single-particle self-energy $\Sigma_{\alpha}(K)$ and the interacting Green’s
function $G_{\alpha}(K)$ referring to helicities $\alpha=\pm$:

$\displaystyle g=\frac{mk_{0}k_{F}}{4\pi}[{1+g\chi(Q)}]t(Q)$ (4)
$\displaystyle\chi(Q)=\frac{2\pi
k_{B}T}{mk_{0}k_{F}\mathcal{V}}\,\sum_{K,\alpha}G_{\alpha}(K)G_{0\,\alpha}(Q-K)w_{\bm{k}-\frac{\bm{q}}{2}}^{2}$
(5) $\displaystyle\Sigma_{\alpha}(K)$
$\displaystyle=\frac{k_{B}T}{\mathcal{V}}\sum_{Q}t(Q)G_{0\,\alpha}(Q-K)w^{2}_{\bm{k}}$
(6) $\displaystyle=G_{0\,\alpha}^{-1}(K)-G_{\alpha}^{-1}(K)$ (7)
$\displaystyle n=\frac{k_{B}T}{\mathcal{V}}\sum_{K,\alpha}G_{\alpha}(K),$ (8)

Here, $n$ is the fermion density. We use the four-momentum
$K\equiv(i\omega_{n},\bm{k})$ and $Q\equiv(i\Omega_{m},\bm{q})$, with
$\omega_{n}=(2n+1)\pi k_{B}T/\hbar$ and $\Omega_{m}=2m\pi k_{B}T/\hbar$ the
Matsubara frequencies for fermion and boson particles, respectively. The
single-particle energy for each $\alpha$-helicity state is
$\epsilon_{\bm{k}}^{\alpha}\equiv\hbar^{2}\bm{k}^{2}/2m-\mu+\alpha\hbar^{2}\lambda
v_{F}|\bm{k}|$, entering the free fermion propagator
$G_{0\,\alpha}=(i\omega_{n}-\epsilon_{\bm{k}}^{\alpha})^{-1}$.

Notice that the approximation expressed by eqs. (4)-(8) is not fully self-
consistent, due to the appearance of the free $G_{0\,\alpha}$ in (5) and (6).
In spite of this simplification, it is known [26, 23] that this approximation
is capable to capture the main spectral properties of the system.

The solution of the set (4)-(8) can be further simplified after resorting to
the analysis developed by Levin et al. [24], embodying all the relevant
physics while being tested by means of full numerical calculations [24]. In
essence, the T-matrix is decomposed into its singular component $t_{sf}$
related to the superfluid part, and a regular component $t_{pg}$ describing
the pseudogap effects arising from the non-condensed pairs, thus

$t(Q)=t_{sf}(Q)+t_{pg}(Q).$ (9)

On the other hand, the superfluid contribution can be approximated to be given
by the BCS expression

$t_{sf}(Q)=-\frac{\Delta_{sf}^{2}}{k_{B}T}\delta(\bm{q})\delta_{i\omega_{n},0},$
(10)

with $\Delta_{sf}$ being the superfluid gap. Expression (10) means that the
fermion-fermion pairing contributes only through the condensate at zero-
momentum. The superfluid self-energy can be expressed as:

$\Sigma_{sf}(K)=\frac{k_{B}T}{\mathcal{V}}\sum_{Q,\alpha}t_{sf}(Q)G_{0\,\alpha}(Q-K)w_{\bm{k}}^{2},$
(11)

so that, eventually, $\Sigma_{sf}$ takes the simple form

$\Sigma_{sf}(K)=-\Delta^{2}_{sf}w_{\bm{k}}^{2}\sum_{\alpha}G_{0\,\alpha}(-K).$
(12)

Inserting (9) and (10) in (4), one obtains, when $Q\neq 0$

$t_{pg}(Q)=\frac{4\pi}{mk_{0}k_{F}}\frac{1}{g^{-1}+\chi(Q)},$ (13)

and, by the same token, the Thouless criterion

$1+g\chi(0)=0,$ (14)

providing the gap equation.

From (6), the splitting of $t(Q)$ leads to a splitting of the self-energy into
$\Sigma(K)=\Sigma_{sf}(K)+\Sigma_{pg}(K)$, with
$\Sigma_{pg}(K)=(k_{B}T/V)\sum_{Q,\alpha}G_{0\,\alpha}(-K+Q)w_{\bm{k}}^{2}t_{pg}(Q)$.
When considering the pseudogap effects in the vicinity of $T_{c}$, one may use
the fact that the T-matrix, $t_{pg}$, is peaked around $Q=0$ [24], and obtain

$\Sigma_{pg}(K)\approx\sum_{\alpha}G_{0\,\alpha}(-K)w_{\bm{k}}^{2}\frac{k_{B}T}{\mathcal{V}}\sum_{Q}t_{pg}(Q).$
(15)

As a result, in analogy to the superfluid gap $\Delta_{sf}$, one may thus
define the pseudogap parameter $\Delta_{pg}$, which in the vicinity of $T_{c}$
takes the form

$\Delta_{pg}^{2}=-\frac{k_{B}T}{\mathcal{V}}\sum_{Q\neq 0}t_{pg}(Q),$ (16)

in terms of the pseudogap T-matrix. The single-particle excitation energy
depending on the helicity $\alpha$ results to be
$E_{\bm{k}}^{\alpha}=\sqrt{(\epsilon_{\bm{k}}^{\alpha})^{2}+\Delta_{\bm{k}}^{2}}$,
with $\Delta_{\bm{k}}\equiv\Delta w_{\bm{k}}$ and
$\Delta^{2}=\Delta_{sf}^{2}+\Delta_{pg}^{2}$ decomposed into its superfluid
and pseudogap parts. The full fermion propagator can eventually be written as:

$G_{\alpha}(K)=\frac{i\hbar\omega_{n}+\epsilon_{\bm{k}}^{\alpha}}{(i\hbar\omega_{n})^{2}-(E_{\bm{k}}^{\alpha})^{2}}.$
(17)

We are now in a position to express the gap and number equations including the
non-condensed pairs. From the Thouless criterion in eq. (14) one obtains:

$1=g\frac{2\pi}{mk_{0}k_{F}\,\mathcal{V}}\sum_{\bm{k},\alpha}\frac{1-2f(E_{\bm{k}}^{\alpha})}{2E_{\bm{k}}^{\alpha}}w_{\bm{k}}^{2},$
(18)

with $f(x)$ the Fermi distribution function. On the other hand, from eq. (8)
one has:

$n=\frac{1}{\mathcal{V}}\sum_{\bm{k},\alpha}\left[\frac{1}{2}\left(1-\frac{\epsilon_{\bm{k}}^{\alpha}}{E_{\bm{k}}^{\alpha}}\right)+\frac{\epsilon_{\bm{k}}^{\alpha}}{E_{\bm{k}}^{\alpha}}f(E_{\bm{k}}^{\alpha})\right].$
(19)

At the critical temperature $T_{c}$, one has $\Delta(T=T_{c})=\Delta_{pg}$.
After expanding the pseudogap T-matrix up to first order in $i\Omega_{m}$ and
second order in $\bm{q}$, one can derive an analytical expression for the
pseudogap parameter [27]. In fact, the expansion coefficients can be expressed
as:

$\begin{split}Z&=\frac{\partial
t^{-1}(Q)}{\partial(i\hbar\Omega_{m})}\Bigg{|}_{Q=0}\\\
\frac{Z}{2m_{b}}&=\frac{\partial^{2}t^{-1}(Q)}{\partial\bm{q}^{2}}\Bigg{|}_{Q=0},\end{split}$
(20)

and measure the renormalized bandwidth $Z$ and effective mass $m_{b}$ of the
non-condensed resonant pairs. In terms of these quantities, one finally has
the expression for the pseudogap $\Delta_{pg}$ [27]:

$\Delta_{pg}^{2}=\frac{1}{Z}\left(\frac{Tm_{b}}{2\pi}\right)^{\frac{3}{2}}\zeta\left(\frac{3}{2}\right),$
(21)

with $\zeta$ the Riemann function.

In the zero temperature limit with $\Delta_{pg}=0$ instead, eqs. (18) and (19)
can be cast in the form:

$1=g\frac{2\pi}{mk_{0}k_{F}\mathcal{V}}\sum_{\bm{k},\alpha}\frac{w_{\bm{k}}^{2}}{2E_{\bm{k}}^{\alpha}},$
(22a)
$n=\frac{1}{2\mathcal{V}}\sum_{\bm{k},\alpha}\left(1-\frac{\epsilon_{\bm{k}}^{\alpha}}{E_{\bm{k}}^{\alpha}}\right).$
(22b)

Once the gap and chemical potential are calculated, the condensate fraction
$N_{c}$ can be worked out from [28]:

$N_{c}=\frac{3\pi^{2}\Delta^{2}}{4\mathcal{V}}\sum_{\bm{k},\alpha}\left(\frac{w_{\bm{k}}}{E_{\bm{k}}^{\alpha}}\right)^{2}.$
(23)

In conclusion, the solution of eqs. (22a), (22b) and (23) provides the
chemical potential $\mu(T=0)$ and superfluid gap $\Delta(T=0)$, along with the
condensate fraction $N_{c}$ at $T=0$. The self-consistent solution of eqs.
(18), (19) and (21), provide the critical temperature $T_{c}$ along with the
pseudogap $\Delta_{pg}(T_{c})$ and chemical potential $\mu(T_{c})$.

##  3 Correlation length

Before discussing the numerical results in the crossover, we need a
quantitative measure of BCS-like and BEC-like behaviors at $T=0$. This is
provided by the correlation length $\xi$, in essence the average width of the
pair-correlation function. In fact, we define the BCS limit of the crossover
as that for which $k_{F}\xi\gg 1$ corresponding to large-size pairs, while the
BEC limit as that with $k_{F}\xi\ll 1$, that is small-size pairs.

In the following, we calculate $\xi$ at $T=0$ in the crossover. To this aim,
we first determine the BCS-like ground state in the very convenient helicity
basis (3). In terms of the indexes $\sigma$ and
$\sigma^{\prime}=\uparrow,\downarrow$, this is:

$|\Theta\rangle=\prod_{\bm{k}}\frac{1+\sum_{\sigma,\sigma^{\prime}}\phi_{\sigma\sigma^{\prime}}(\bm{k})c_{\bm{k}\sigma}^{\dagger}c_{-\bm{k}\sigma^{\prime}}^{\dagger}}{\sqrt{1+\sum_{\sigma\sigma^{\prime}}|\phi_{\sigma\sigma^{\prime}}(\bm{k})|^{2}}}|0\rangle,$
(24)

where

$\begin{split}&\phi_{\uparrow\downarrow}(\bm{k})=-\phi_{\downarrow\uparrow}(\bm{k})=-\frac{1}{2}\Big{(}\phi_{\bm{k}}^{+}s_{+}+\phi_{\bm{k}}^{-}s_{-}\Big{)}\frac{k_{\perp}}{|\bm{k}|},\\\
&\phi_{\uparrow\uparrow}(\bm{k})=-\phi_{\downarrow\downarrow}^{\ast}(\bm{k})=\frac{1}{2}\Big{(}\phi_{\bm{k}}^{+}-\phi_{\bm{k}}^{-}\Big{)}\frac{k_{\perp}}{|\bm{k}|}e^{-i\varphi_{\bm{k}}}.\end{split}$

Here, $\varphi_{\bm{k}}\equiv arg(k_{x}+ik_{y})$, as defined before, we have
defined $\phi_{\bm{k}}^{\alpha}=v_{\bm{k}}^{\alpha}/u_{\bm{k}}^{\alpha}$ and

$(u_{\bm{k}}^{\alpha})^{2}=\frac{1}{2}\left(1+\frac{\epsilon_{\bm{k}}^{\alpha}}{E_{\bm{k}}^{\alpha}}\right)\,\,\,\text{and}\,\,\,(v_{\bm{k}}^{\alpha})^{2}=\frac{1}{2}\left(1-\frac{\epsilon_{\bm{k}}^{\alpha}}{E_{\bm{k}}^{\alpha}}\right)$

playing the role of the BCS $u_{\bm{k}}$ and $v_{\bm{k}}$ in the helicity
basis labeled by $\alpha$.

Now, we may introduce the pair distribution function:

$g_{\sigma\sigma^{\prime}}(\bm{r}-\bm{r}^{\prime})=\frac{1}{n^{2}}\langle\Theta|[\hat{\rho}_{\sigma}(\bm{r})\hat{\rho}_{\sigma^{\prime}}(\bm{r^{\prime}})-\delta_{\sigma\sigma^{\prime}}\delta(\bm{r}-\bm{r}^{\prime})\hat{\rho}_{\sigma}(\bm{r})]|\Theta\rangle,$

normalized to the squared average density $n$ of the system, with
$\hat{\rho}_{\sigma}(\bm{r})=c_{\sigma}^{\dagger}(\bm{r})c_{\sigma}(\bm{r})$
being the density operator for particles with spin $s$ and position $\bm{r}$.
At variance with the conventional BCS case without SOC, here the pair-
correlation function is in general characterized by non-zero amplitudes in
both the singlet and triplet channels. Thus, we define
$g(\bm{r})\equiv\sum_{\sigma\sigma^{\prime}}g_{\sigma\sigma^{\prime}}(\bm{r})$
and the corresponding (squared) pair correlation length as
$\xi^{2}=[\int\bm{r}^{2}g(\bm{r})d\bm{r}]/[\int g(\bm{r})d\bm{r}]$. After
Fourier transformation and neglecting - as it is customary - Hartree and Fock
contributions [29], we obtain the expression:

$\xi^{2}=\frac{1}{\mathcal{N}}\int
d\bm{k}\sum_{\sigma,\sigma^{\prime}}\left|\nabla_{\bm{k}}A_{\bm{k}}^{\sigma,\sigma^{\prime}}\right|^{2},$
(25)

with $\mathcal{N}\equiv{\int
d\bm{k}\sum_{\sigma,\sigma^{\prime}}\left|A_{\bm{k}}^{\sigma,\sigma^{\prime}}\right|^{2}}$.
Eq. (25) generalizes the BCS expression for $\xi$ [7] to the inclusion of SOC,
with:
$A_{\bm{k}}^{\sigma,\sigma^{\prime}}\equiv{\phi_{\sigma,\sigma^{\prime}}(\bm{k})}[1+\sum_{\sigma\sigma^{\prime}}|\phi_{\sigma\sigma^{\prime}}(\bm{k})|^{2}]^{-1}$.
In the following, we discuss the behavior of the $T=0$ (from eqs. (22a), (22b)
and (23)) and $T=T_{c}$ (from eqs (18), (19) and (21)) quantities in the
crossover spanned by $k_{F}\xi$ (as calculated from (25)), while varying the
strength $g$ and range $k_{0}^{-1}$ of the pairing interaction, and the SOC
strength $\lambda$. To this aim, we first analyze the behavior of the pair
correlation length and of the threshold of $\lambda$ above which the change of
topology occurs in the Fermi sphere.

##  4 Correlation length and threshold SOC strength: competition between SOC
and finite-range effects

Fig. 1 displays the calculated correlation length $k_{F}\xi$ as a function of
$g$ for different values of the SOC strength $\lambda$ (different colours as
in the legend) and $k_{0}$ (different line types). We notice that, at fixed
$g$, the introduction of SOC (yellow lines) favors a BEC-like regime with
smaller pair sizes with respect to the absence of SOC (blue lines). Let us
comment on the effect of finite interaction range. In the absence of SOC,
longer interaction ranges (dotted-dashed lines) on the scale of $k_{F}^{-1}$,
are seen to favor a BCS-like regime with larger pair sizes while $g>1$. For
weaker coupling values $g<1$ than the threshold $g\approx 1$, the behavior is
inverted and shorter ranges correspond to more BCS-like character. Once the
SOC is switched on, the threshold of $g$ where the inverted behavior occurs,
is lowered towards weaker coupling strengths, confirming the role of SOC in
extending the BEC region of the crossover.

Figure 1: Pair correlation length $k_{F}\xi$ vs. the coupling $g$, without
(blue lines) and with (yellow lines) SOC, for different interaction ranges as
in the legend: $k_{0}=0.5$ (dotted line), $k_{0}=1$ (dotted-dashed line) and
$k_{0}=4$ (solid line). The horizontal dashed lines $k_{F}\xi=2\pi$ and
$k_{F}\xi=\pi^{-1}$ correspond to the thresholds for deep BCS-like
($k_{F}\xi>2\pi$) or deep BEC ($k_{F}\xi<\pi^{-1}$) behavior in the crossover
without SOC [7].

From Fig. 1, we can argue that in the presence of stronger attractive coupling
$g$, the formation of larger size fermion pairs is favored by the action of
longer rather than shorter interaction ranges, and viceversa for weaker
coupling values of $g$. When SOC is switched on, the behavior at weaker
coupling strengths is evidently flattened towards a BEC-like regime. As
expected from eq. (24), SOC is seen to favor the emergence of triplet pairing
with amplitudes $\phi_{\uparrow\uparrow}$ and $\phi_{\downarrow\downarrow}$,
confirming the predictions from Ref. [21]. In addition, we find that the
triplet amplitude is enhanced in the limit of weak coupling strengths in
accord with [21] and, consistently with the effect of $k_{0}$ emerged so far,
also for longer ranges of the interactions. The second parameter of the theory
is the SOC strength. One question is therefore to which extent the threshold
for the change of topology in the Fermi sphere is affected by the
interactions. As detailed in Fig. 2 of the Supplemental material [30], we find
that $\lambda_{B}$ is unchanged while in the deep BCS regime with
$k_{F}\xi\gtrsim 2\pi$, independently of the interaction range, while the
threshold $\lambda_{B}$ tends to be slightly lowered in the intermediate
crossover region before entering the deep BEC limit. Longer-range interactions
tend to contrast the effect of SOC in the stronger coupling regime with
smaller $k_{F}\xi$, so that for the case with $k_{0}=0.5$, $\lambda_{B}$ is
enhanced with respect to the ideal system in a small region just below
$k_{F}\xi=2\pi$.

In the following, the behavior of $\xi$ and $\lambda_{B}$ is used to frame the
parameters region for which universal behavior occurs in the crossover.

##  5 Universality

We now turn to discuss the behavior of the observables at $T=0$ and $T=T_{c}$.
We first notice that the observed competition behavior is found also while
analyzing the results for the superfluid gap $\Delta$ and condensate fraction
$N_{c}$ at $T=0$ as functions of $\lambda$ for different values of $k_{0}$ in
the weak-coupling regime with $g<1$. Larger values of $\Delta$ and $N_{c}$
expected for BEC-like behavior, characterize the system with larger SOC and
shorter-range interactions at fixed $g$, while for weak SOC strengths $N_{c}$
and $\Delta$ have smaller values.

In fact, the dependence of the condensate fraction, chemical potential, and
critical temperature, on the different parameters appears to be greatly
simpliflied once it is analyzed in terms of the correlation length $k_{F}\xi$.
The degree of simplification is striking when compared with the behavior of
the same quantities in terms of $g$, $k_{0}$, and $\lambda$ separately, as
displayed in the Supplemental material [30]. This is displayed in Fig. 2,
where all the data sets corresponding to different values of $g$, $k_{0}$ and
$\lambda$, are collected and shown vs. the corresponding $k_{F}\xi$. A
striking universal behavior is observed for weak SOC strengths
$\lambda\leq\lambda_{B}$, crossing over from the BEC regime with small
$k_{F}\xi<\pi^{-1}$ to the BCS regime with large $k_{F}\xi>2\pi$. This
universal behavior is however lost - not surprisingly - while
$\lambda>\lambda_{B}$ and the Fermi sphere is changing its topology: here, the
formation of the BEC is seen to be enhanced at a faster rate than for smaller
SOC strengths, as expected.

Figure 2: Universal behaviour of the condensate fraction $N_{c}$ at $T=0$ vs.
the pair correlation length $k_{F}\xi$, varied after changing $g$ with $k_{0}$
and $\lambda$, as in the legend. Given the value of $\lambda_{B}$, in terms of
the correlation length in Fig. 2 of the Supplemental material [30]. For
$\lambda<\lambda_{B}$, the values of $k_{0}$ are represented by different
symbols: $k_{0}=0.5$ (squares), $k_{0}=1$ (circles) and $k_{0}=4$ (triangles).
For $\lambda>\lambda_{B}$, the values of $k_{0}$ are represented by different
line types: $k_{0}=0.5$ (dotted line), $k_{0}=1$ (dashed line) and $k_{0}=4$
(solid line). The coupling $g$ is in the range $0.05<g<50$. The vertical
dashed lines at $k_{F}\xi=\pi^{-1}$ and $k_{F}\xi=2\pi$ signal the deep BEC
and BCS thresholds, respectively. The value of $\lambda_{B}$ is obtained from
Fig. 2 in the Supplemental material [30]

As to the chemical potential, the universal behavior is better seen after
definining the effective quantity $\tilde{\mu}=\mu+m(\lambda v_{F})^{2}/2$,
i.e. the chemical potential measured from the bottom of the energy of the
helicity state with lower energy.

Figure 3: Universal behaviour of the effective chemical potential
$\tilde{\mu}$ (see text) vs. the pair correlation length $k_{F}\xi$, varied
after changing $g$ with $k_{0}$ and $\lambda$, as in the legend.

For $\tilde{\mu}<0$ on the BEC side, $\tilde{\mu}$ is normalized to the
limiting value $\tilde{\mu}_{BEC}$, that is half the pair binding energy. For
$\tilde{\mu}>0$, $\tilde{\mu}$ is normalized to its non-interacting limiting
value $\tilde{\mu}_{NI}$ (see text). Symbols are as in Figure 2. The coupling
$g$ is in the range $0.05<g<50$. For $\lambda>\lambda_{B}$ the universal
behavior is lost: here, the case with $\lambda=1.5$ and $k_{0}=4$ is shown for
comparison. The vertical dashed lines at $k_{F}\xi=\pi^{-1}$ and
$k_{F}\xi=2\pi$ signal the deep BEC (small pair sizes) and BCS (large pair
sizes) thresholds, respectively. Inset: Zoom into the region with
$10.6<k_{F}\xi<12.1$. Difference between $\tilde{\mu}$ and its corresponding
non-interacting value $\tilde{\mu}_{NI}$, as a function of $\lambda$: for
large $\lambda$, pair correlation lengths $k_{F}\xi>2\pi$ do not correspond to
a weakly interacting BCS regime, independently of the interaction range (here,
$k_{0}=4$).

The effective chemical potential $\tilde{\mu}$ is displayed in Fig. 3 vs.
$k_{F}\xi$, varied after changing $g$, $k_{0}$ and $\lambda$, as in the
legend. When $\tilde{\mu}<0$, $\tilde{\mu}$ is normalized to the limiting
value $\tilde{\mu}_{BEC}$, that is half the pair binding energy. This is
obtained after numerical solution of the expression
$1=gk_{0}[c^{2}(1+k_{0})+\lambda^{2}]/\\{(16c)[(c+k_{0})^{2}+\lambda^{2}]\\}$,
where $c^{2}=-\tilde{\mu}/E_{F}$. When $\tilde{\mu}>0$, $\tilde{\mu}$ is
normalized to the non-interacting limiting value $\tilde{\mu}_{NI}$ given by
$\tilde{\mu}_{NI}/{E_{F}}=2^{{1}/{3}}\lambda^{2}/F(\lambda)+(F(\lambda)/2)^{1/3}-2\lambda^{2}$,
with $F(\lambda)\equiv 1+2\lambda^{6}+\sqrt{1+4\lambda^{6}}$ [31]. A striking
universal behavior persists while $\lambda$ is below the threshold
$\lambda_{B}$ for the topological change of the Fermi sphere, extending the
results in [7] to the inclusion of weak SOC interactions. Once again however,
the data for $\lambda>\lambda_{B}$ fail to fall in the universal curve,
essentially because $\tilde{\mu}$ does not stick to the BCS-like non-
interacting limit. As it often occurs, the failure of universal behavior is
far more evident in the chemical potential than in the condensate fraction.
This dramatic change in behavior is zoomed in the inset of Fig. 3, where the
difference between the effective chemical potential and its non-interacting
counterpart is displayed for values of $k_{F}\xi$ well within the deep weakly-
interacting BCS regime as a function of $\lambda$. In fact, $\tilde{\mu}$ is
seen to significantly depart from $\tilde{\mu}_{NI}$. We thus observe that,
though the pairs are on average characterized to be of large size, the weakly-
interacting BCS state is not any-longer occurring well above the topological
change. As a result, we infer that we do not have a crossover from a BCS to a
BEC state but more properly from an effective BCS state of interacting bosonic
molecules made of fermions with same helicity to an usual BEC state,
confirming the results obtained by Yamaguchi et al. [32].

Finally, we show in Fig. 4 the critical temperature $T_{c}m_{b}$ as a function
of $k_{F}\xi$. Again, a striking universal behavior persists as long as
$\lambda<\lambda_{B}$. Notice that the collapse of all the data with different
values of $g$, $k_{0}$, and $\lambda$ occurs only after embodying the
effective-mass correction arising for the non-condensed resonant pairs in eq.
(20). The data sets with $\lambda>\lambda_{B}$ are seen to depart from the
universal curve, to an extent similar to the behavior of $N_{c}$ rather than
that of $\tilde{\mu}$.

Figure 4: Universal behaviour of the critical temperature $T_{c}$ times the
effective mass $m_{b}$ of the non-condensed resonant pairs as a function of
the pair correlation length $k_{F}\xi$ in log scale, for different values of
$\lambda$ as in the legend. Symbols refer to different values of $k_{0}=0.5$
(squares) and $k_{0}=4$ (triangles). The coupling $g$ is in the range
$1.25<g<30$. The vertical dashed lines at $k_{F}\xi=\pi^{-1}$ and
$k_{F}\xi=2\pi$ signal the deep BEC and BCS thresholds, respectively.

##  6 Conclusions

We have studied the effects of light-induced spin-orbit coupling (SOC) on the
crossover from a Bardeen-Cooper-Schrieffer (BCS) to a Bose-Einstein
Condensation (BEC) type of superfluidity in an interacting quantum gas of
fermionic atoms in two different spin states, including at mean-field level
the effects of the non-condensed pairs via the pairing approximation developed
by [22, 24]. We have focused on the so far unexplored effects obtained by
tuning the range of the inter-atomic interactions and the degree of universal
behaviour in the crossover.

In particular, we find that in strong coupling conditions, a competition sets
in between the SOC interaction, which moves the system towards the BEC side,
and longer interaction ranges, which move the system towards the BCS side. At
weak coupling, stronger SOC and longer range interactions cooperate to move
the system towards the BEC side. In addition, we observe that universal
behavior persists in the condensate fraction $N_{c}$ and effective chemical
potential $\tilde{\mu}$ at $T=0$, as well as in the critical temperature
$T_{c}$ while remaining below the topological transition. In order to observe
universality, we had to use the correlation length as a driving parameter,
measure the chemical potential from the bottom of the lowest helicity state,
and embody effective-mass renormalization effects of the resonant non-
condensed pairs into the critical temperature. Above the topological
transition instead, universality is progressively lost. We may argue that the
standard BCS-BEC crossover could be replaced by a crossover from a BEC of
tightly-bound composite bosons to a BCS of interacting bosonic molecules
composed of same-helicity quasi-particles [21]. In fact, we can speculate that
a calculation of the effective mass or size of the fermion pairs while moving
away from the BEC regime, can be used to upgrade the universal parameter $\xi$
with inclusion of a new length which might recover a form of universal
behavior for stronger SOC.

Our predictions can be tested in timely experiments with, e.g., effective
photon-mediated interactions in optical cavities [14], and can be interesting
in view of applications exploiting topological states as they can be induced
by SOC.

## References

  * [1] Bardeen J., Cooper L. N. Schrieffer J. R., Phys. Rev. 108 (1969) 1175.
  * [2] Strinati G. C., Pieri P., Röpke G., Schuck P. Urban M., Phys. Rep. 738 (2018) 1.
  * [3] Leggett A. J., in Modern Trends in the Theory of Condensed Matter, edited by Pekalski A. and Przystawa J., Vol.115 (springer-Verlag) 1980, pp.13-27.
  * [4] Nozières P. Schmitt-Rink S., J. Low Temp. Phys. 59 (1985) 195-211.
  * [5] Uemura Y. J., Le L. P. , Luke G. M. , Sternlieb B. J., Wu W. D., Brewer J. H., Riseman T. M., Seaman C. L., Maple M. B., Ishikawa M. , Hinks D. G., Jorgensen J. D., Saito G. Yamochi H., Phys. Rev. Lett. 66 (1991) 2665.
  * [6] Randeria M., in Bose-Einstein condensation, edited by Griffin A., Snoke D. W. Stringari S. (Cambridge University Press) 1995, pp. 355-392.
  * [7] Pistolesi F. Strinati G. C., Phys. Rev. B 49 (1994) 6356.
  * [8] Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E., Cornell E. A., Science 269 (1995) 5221.
  * [9] Davis K. B., Mewes M.-O., Andrews M. R., van Druten N. J., Durfee D. S., Kurn D. M. Ketterle W., Phys. Rev. Lett. 75 (1995) 3969.
  * [10] Greiner M., Regal C. A. Jin D. S., Nature 426 (2003) 537.
  * [11] Zwierlein M. W., Stan C. A., Schunck C. H., F. Raupach S. M. , Gupta S., Hadzibabic Z. Ketterle W., Phys. Rev. Lett. 91 (2003) 250401.
  * [12] Fano U., Phys. Rev. 124 (1961) 1866; Feshbach H., Ann. Phys. 19 (1962) 287.
  * [13] For a review, see: Chin C., Grimm R., Julienne P., Tiesinga E., Rev. Mod. Phys. 82 (2010) 1225 and references therein.
  * [14] Vaidya V. D., Guo Y., Kroeze R. M., Ballantine K. E., Kollár A. J., Keeling J. Lev B. L., Phys. Rev. X 8 (2018) 011002.
  * [15] Dalibard J., Gerbier F., Juzeliūnas G. Öhberg P., Rev. Mod. Phys. 83 (2011) 1523.
  * [16] Lin Y.-J., Compton R. L., Jiménez-García K., Porto J. V. Spielman I. B., Nature 462 (2009) 628; Lin Y.-J., Compton R. L., Perry A. R., Phillips W. D., Porto J. V. Spielman I. B.. Phys. Rev. Lett. 102 (2009) 130401.
  * [17] Wu Z., Zhang L., Sun W., Xu X.-T., Wang B.-Z., Ji S.-C., Deng Y., Chen S., Science 354 (2016) 83.
  * [18] Wang P., Yu Z.-Q., Fu Z., Miao J., Huang L., Chai S., Zhai H. Zhang J., Phys. Rev. Lett. 109 (2012) 095301.
  * [19] Huang L., Meng Z., Wang P., Peng P., Zhang S.-L., Chen L., Li D., Zhou Q. Zhang J., Nature Physics 12 (2016) 540–544.
  * [20] Zhai H., Rep. Prog. Phys. 78 (2015) 026001.
  * [21] Vyasanakere J. P. Shenoy V. B., Phys. Rev. B 83 (2011) 094515; Vyasanakere J. P., Zhang S. Shenoy V. B., Phys. Rev. B 84 (2011) 014512; Shenoy V. B., Vyasanakere J. P. Ghosh S. K., Current Science 103 (2012) 525-535.
  * [22] Kadanoff L. P. Martin P. C., Phys. Rev. 124 (1961) 670.
  * [23] Šopík B., Lipavský P., Männel M., Morawetz K., Matlock P., Phys. Rev. B 84 (2011) 094529.
  * [24] Jankó B., Maly J. Levin K.,Phys. Rev. B 56 (1997) R11407; Kosztin I., Chen Q., Jankó B. Levin K., Phys. Rev. B 58 (1998) R5936; Chen Q., Kosztin I., Jankó B. Levin K., Phys. Rev. Lett.81 (1998) 4708.
  * [25] Wu C.-T., Anderson B. M., Boyack R. Levin K., Phys. Rev. B 91 (2015) R220504.
  * [26] Stefanucci G. van Leeuwen R., Nonequilibrium Many-Body Theory of Quantum Systems (Cambridge University Press) 2013.
  * [27] He L., Huang X.-G., Hu H. Liu X.-J., Phys. Rev. A87 (2013) 053616.
  * [28] He L. Huang. X.-G., Phys. Rev. B 86 (2012) 014511.
  * [29] Strinati G. C., in Models and Phenomenology for Conventional and High-temperature Superconductivity, edited by Iadonisi G., Schrieffer J. R. Chiofalo M. L. (IOS Press) 1998, pp. 7-51.
  * [30] See supplemental material for details on the dependence of $\lambda_{B}$ on the interactions and for the behavior of $N_{c}$, $\mu$, and $T_{c}$ on the parameters $g$, $k_{0}$, and $\lambda$ separately.
  * [31] He L. Yu Z.-Q., Phys. Rev. A 84 (2011) 025601.
  * [32] Yamaguchi T., Inotani D. Ohashi Y., J. Low Temp. Phys. 183 (2016) 161-168.


# Spectral mapping theorem and the Taylor spectrum

Muneo Chō, B. Načevska Nastovska and Kôtarô Tanahashi

###### Abstract

In [6] Chō and Tanahashi showed new spectral mapping theorem of the Taylor
spectrum for doubly commuting pairs of $p$-hyponormal operators and
$\log$-hyponormal operators. In this paper, we will show that same spectral
mapping theorem holds for commuting $n$-tuples.

00footnotetext: 2010 Mathematics Subject Classification; Primary 47B20,
Secondary 47A10.Keywords; Helbert space, Taylor spectrum, spectral mapping
theorem, $p$-hyponormal, $\log$-hyponormal.

##  1 Introduction and preparation

Let $\mathcal{H}$ be a complex Hilbert space and $B(\mathcal{H})$ be the set
of all bounded linear operators on $\mathcal{H}$. For $T\in B(\mathcal{H})$,
let $\sigma(T),\sigma_{p}(T)$ and $\sigma_{a}(T)$ denote the spectrum, the
point spectrum and the approximate point spectrum of $T$, respectively. Let
$\lambda\in\mathbb{C}$ belong to the residual spectrum $\sigma_{r}(T)$ of $T$
if there exists $c>0$ such that $\|(T-\lambda)x\|\geq c\|x\|$ for all
$x\in{\mathcal{H}}$ and $(T-\lambda){\mathcal{H}}\not={\mathcal{H}}$. It is
easy to see that if $\lambda\in\sigma_{r}(T)$, then
$0\in\sigma_{p}((T-\lambda)^{*})$. It is well known that
$\sigma(T)=\sigma_{a}(T)\cup\sigma_{r}(T)$. For an Hermitian operator $A\in
B(\mathcal{H})$, we denote $A\geq 0$ if $(Ax,x)\geq 0$ for every
$x\in\mathcal{H}$ and $A\geq B$ if $A-B\geq 0$. When $(Ax,x)>0$ for every non-
zero $x\in\mathcal{H}$, then we denote $T>0$. For a given $p>0$, $T\in
B(\mathcal{H})$ is said to be $p$-hyponormal if
$(T^{*}T)^{p}\geq(TT^{*})^{p}$. When $p=1/2$, $T$ is said to be semi-
hyponormal. It means that $T$ is semi-hyponormal if and only if
$|T|\geq|T^{*}|$. $T$ is said to be $\log$-hyponormal if $T$ is invertible and
$\log|T|\geq\log|T^{*}|$. It is well known that if $T$ is invertible
$p$-hyponormal for some $p>0$, then $T$ is $\log$-hyponormal. If
${\mathcal{M}}$ is a reducing subspace for a $p$-hyponormal or
$\log$-hyponormal operator $T$, then so is $T|_{{\mathcal{M}}}$, respectively.

For a commuting $n$-tuple ${\bf T}=(T_{1},...,T_{n})\in B(\mathcal{H})^{n}$,
we explain the Taylor spectrum $\sigma({\bf T})$ of ${\bf T}$ shortly. Let
$E^{n}$ be the exterior algebra on $n$ generators, that is, $E^{n}$ is the
complex algebra with identity $e$ generated by indeterminates
$e_{1},...,e_{n}$. Let $E^{n}_{k}(\mathcal{H})=\mathcal{H}\otimes E^{n}_{k}$.
Define $D^{n}_{k}\ :\ E^{n}_{k}(\mathcal{H})\ \longrightarrow\
E^{n}_{k-1}(\mathcal{H})$ by

$D^{n}_{k}(x\otimes e_{j_{1}}\wedge\cdots\wedge
e_{j_{k}}):=\sum_{i=1}^{k}(-1)^{i-1}T_{j_{i}}x\otimes
e_{j_{1}}\wedge\cdots\wedge\check{e}_{j_{i}}\wedge\cdots\wedge e_{j_{k}},$

where $\check{e}_{j_{i}}$ means deletion. We denote $D^{n}_{k}$ by $D_{k}$
simply. We think Koszul complex $E({\bf T})$ of ${\bf T}$ as follows:

$\ \ \ \ E({\bf T})\ :\ 0\longrightarrow
E^{n}_{n}(\mathcal{H})\stackrel{{\scriptstyle
D_{n}}}{{\longrightarrow}}E^{n}_{n-1}(\mathcal{H})\ \stackrel{{\scriptstyle
D_{n-1}}}{{\longrightarrow}}\ \cdots\stackrel{{\scriptstyle
D_{2}}}{{\longrightarrow}}E^{n}_{1}(\mathcal{H})\stackrel{{\scriptstyle
D_{1}}}{{\longrightarrow}}E^{n}_{0}(\mathcal{H})\longrightarrow 0.$

Since $E^{n}_{k}(\mathcal{H})\ \cong\
\overbrace{\mathcal{H}\oplus\cdots\oplus\mathcal{H}}^{\binom{n}{k}=\frac{n!}{(n-k)!\,k!}}\
\ (k=1,...,n),$ we set $E^{n}_{k}(\mathcal{H})\ =\
\overbrace{\mathcal{H}\oplus\cdots\oplus\mathcal{H}}^{\binom{n}{k}}\ \
(k=1,...,n).$

######  Definition 1.1.

A commuting $n$-tuple ${\bf T}=(T_{1},...,T_{n})\in B(\mathcal{H})^{n}$ is
said to be singular if and only if the Koszul complex $E({\bf T})$ of ${\bf
T}$ is not exact.

######  Definition 1.2.

For a commuting $n$-tuple ${\bf T}=(T_{1},...,T_{n})\in B(\mathcal{H})^{n}$,
$z=(z_{1},...,z_{n})\in\mathbb{C}^{n}$ belongs to the Taylor spectrum
$\sigma_{T}({\bf T})$ of $\bf{T}$ if ${\bf T}-z=(T_{1}-z_{1},...,T_{n}-z_{n})$
is singular.

About the definition of the Taylor spectrum, see details J. L. Taylor [9] and
[10]. In [7], Curto proved the following proposition.

######  Proposition  1.3.

(Proposition 3.4, Curto [7])  For a commuting $n$-tuple ${\bf
T}=(T_{1},...,T_{n})\in B(\mathcal{H})^{n}$,
$0=(0,...,0)\not\in\sigma_{T}({\bf T})$ if and only if
$D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*}$ is invertible for all $k$.

For a commuting pair ${\bf T}=(T_{1},...,T_{n})\in B(\mathcal{H})^{n}$, it is
well known that, for polynomials $f_{1},...,f_{m}$ of $n$-variables, if
$f(z_{1},...,z_{n})=(f_{1}(z_{1},...,z_{n}),...,f_{m}(z_{1},...,z_{n}))$, then
it holds

$\sigma_{T}(f(T_{1},...,T_{n}))=f(\sigma_{T}(T_{1},...,T_{n})),$

where $\sigma_{T}(T_{1},...,T_{n})$ is the Taylor spectrum of ${\bf
T}=(T_{1},...,T_{n}).$ See Theorem 4.7 in [10].

In the paper [6], Chō and Tanahashi showed another spectral mapping theorem
under the following assumption.

Let $T=U|T|\in B(\mathcal{H})$ be the polar decomposition of $T$ with unitary
$U$ and $f$ be a continuous function on the non-negative real line which
contains $\sigma(|T|)$. Let ${\mathcal{K}}$ be Berberian extension of
$\mathcal{H}$ and $\circ:B(\mathcal{H})\ni T\rightarrow T^{\circ}\in
B({\mathcal{K}})$ be a faithful $*$-representation. We set the following
conditions (1) and (2):

(1) $\displaystyle\text{For a sequence }\\{x_{n}\\}\text{ of unit vectors, if
}\,(T-z)x_{n}\rightarrow 0,\ \text{then}\ (T-z)^{*}x_{n}\rightarrow 0.$ (2)
$\displaystyle\text{If a closed subspace }\ {\mathcal{M}}\ \text{ of }\
{\mathcal{K}}\ \text{reduces }\ T^{\circ}\ \text{ and}\
re^{i\theta}\in\sigma(T^{\circ}|_{\mathcal{M}}),$ $\displaystyle\qquad\text{
then }\ {\mathcal{M}}\ \text{ reduces }\ U^{\circ},|T|^{\circ}\ \text{and}\
e^{-i\theta}f(r)\in\sigma_{p}\left((U^{\circ}|_{\mathcal{M}}f(|T|^{\circ})|_{\mathcal{M}})^{*}\right).$

######  Theorem 1.4.

Let ${\bf T}=(T_{1},T_{2})$ be a doubly commuting pair of operators and
$T_{j}=U_{j}|T_{j}|\ (j=1,2)$ be the polar decomposition. Let $f(t)$ be a
continuous function on a open interval in the non-negative real line which
contains $\sigma(|T_{1}|)\cup\sigma(|T_{2}|)$. Let $S_{j}=U_{j}f(|T_{j}|)\
(j=1,2)$ and ${\bf S}=(S_{1},S_{2})$. Let $T_{1},T_{2}$ and $f$ satisfy $(1)$
and $(2)$. If $(r_{1}e^{i\theta_{1}},r_{2}e^{i\theta_{2}})\in\sigma_{T}({\bf
T}),\ \mbox{then }\
(e^{i\theta_{1}}f(r_{1}),e^{i\theta_{2}}f(r_{2}))\in\sigma_{T}({\bf S}).$

See the details of Berberian extension [1]. That proof depends on the
following Vasilescu’s result.

Let ${\bf T}=(T_{1},T_{2})$ be a commuting pair of operators on $\mathcal{H}$,
${\bf z}=(z_{1},z_{2})\in\mathbb{C}^{2}$ and let

$\alpha({\bf T}-{\bf z}):=\left(\begin{array}[]{cc}T_{1}-z_{1}&T_{2}-z_{2}\\\
-(T_{2}-z_{2})^{*}&(T_{1}-z_{1})^{*}\end{array}\right)\ \ \mbox{on }\
\mathcal{H}\oplus\mathcal{H}.$

Then Vasilescu proved the following result.

######  Proposition  1.5.

(Theorem 1.1, Vasilescu [11])  Let ${\bf T}=(T_{1},T_{2})\in
B(\mathcal{H})^{2}$ be a commuting pair. Then

${\bf z}=(z_{1},z_{2})\in\sigma_{T}({\bf T})\ \mbox{if and only if }\
\alpha({\bf T}-{\bf z})\ \mbox{is not invertible}.$

Therefore, we have ${\bf z}=(z_{1},z_{2})\in\sigma_{T}({\bf T})\ \ \mbox{if
and only if }\ \ 0\in\sigma(\alpha({\bf T}-{\bf z})).$

For an $n$-tuple ${\bf T}=(T_{1},...,T_{n})$, the joint point spectrum
$\sigma_{jp}({\bf T})$ is the set of all numbers ${\bf
z}=(z_{1},...,z_{n})\in\mathbb{C}^{n}$ such that there exists a non-zero
vector $x\in\mathcal{H}$ which satisfies $T_{j}x=z_{j}x\ (\forall j=1,...,n)$
and the joint approximate point spectrum $\sigma_{ja}({\bf T})$ is the set of
all numbers ${\bf z}=(z_{1},...,z_{n})\in\mathbb{C}^{n}$ such that there
exists a sequence $\\{x_{k}\\}$ of unit vectors of $\mathcal{H}$ which
satisfies

$(T_{j}-z_{j})x_{k}\ \to\ 0\ \mbox{as}\ k\to\infty\ (\forall j=1,...,n).$

Following proposition is due to Berberian [1] for a single operator case. It
is easy to see a proof for $n$-tuples. See Berberian [1] and Chō [2].

######  Proposition  1.6.

Let $B(\mathcal{H})$ be the set of all bounded linear operators on
$\mathcal{H}$. Then there exist an extension space ${\mathcal{K}}$ of
$\mathcal{H}$ and a faithful $*$-representation of $B(\mathcal{H})$ into
$B({\mathcal{K}}):\ T\ \to\ T^{\circ}$ such that

$\sigma_{ja}({\bf T})=\sigma_{ja}({\bf T}^{\circ})=\sigma_{jp}({\bf
T}^{\circ}),$

where ${\bf T}=(T_{1},...,T_{n})\in B(\mathcal{H})^{n}$ and ${\bf
T}^{\circ}=(T_{1}^{\circ},...,T_{n}^{\circ})$.

Following results are well known.

######  Proposition  1.7.

Let $T=U|T|$ be the polar decomposition of $T$ and $f$ be a continuous
function on the non-negative real line which contains $\sigma(|T|)$. For a
sequence $\\{x_{n}\\}$ of unit vectors, if $(T-re^{i\theta})x_{n}\rightarrow
0$ and $(T-re^{i\theta})^{*}x_{n}\rightarrow 0$, then
$(U-e^{i\theta})x_{n}\rightarrow 0,(|T|-r)x_{n}\rightarrow 0$ and
$(f(|T|)-f(r))x_{n}\rightarrow 0$.

See Lemma 1.2.4 in [13].

######  Proposition  1.8.

Let $T$ be semi-hyponormal. Then
$\sigma(T)=\\{\overline{z}:z\in\sigma_{a}(T^{*})\\}$.

See Theorem 1.2.6 in [13].

Remark. If $T$ is $p$-hyponormal and $f(t)=t^{2p}$, then (2) holds by Theorem
4 of [3]. If $T$ is $\log$-hyponormal and $f(t)=\log\,t$, then (2) holds by
Lemma 3 of [8]. About (3), since the mapping $\circ$ of Berberian method is a
faithful $*$-representation, so is $T^{\circ}$ if $T$ is $p$-hyponormal or
$\log$-hyponormal, respectively. Let ${\mathcal{M}}$ be a reducing subspace
for $T$. It is clear that if $T$ is $p$-hyponormal or $\log$-hyponormal, then
so is $T|_{{\mathcal{M}}}$, respectively.
(i) Let $T$ be $p$-hyponormal and $T=U|T|$ be the polar decomposition of $T$
and $f(t)=t^{2p}$. Then $S=U|T|^{2p}$ is semi-hyponormal and
$\sigma(U|T|^{2p})=\\{r^{2p}e^{i\theta}\,:\,re^{i\theta}\in\sigma(T)\,\\}$ by
Theorem 3 of [4]. Hence (3) holds by Proposition 1.8.
(ii) Let $T=U|T|$ be $\log$-hyponormal and $f(t)=\log\,t$. Then $S=U\log|T|$
is semi-hyponormal and
$\sigma(U\log|T|)=\\{e^{i\theta}\log\,r\,:\,re^{i\theta}\in\sigma(T)\,\\}$ by
Lemma 8 of [8]. Hence (3) holds by Proposition 1.8.
Therefore, if $T$ is $p$-hyponormal or $\log$-hyponormal and $f(t)=t^{2p}$ or
$f(t)=\log\,t,$ respectively, then $T$ satisfies (2) and (3) for this $f$.

In this paper, we would like to prove the following theorem.

######  Theorem 1.9.

Let ${\bf T}=(T_{1},...,T_{n})$ be a doubly commuting $n$-tuple of operators
and $T_{j}=U_{j}|T_{j}|\ (j=1,...,n)$ be the polar decompositions. Let $f(t)$
be a continuous function on a open interval in the non-negative real line
which contains $\sigma(|T_{1}|)\cup\cdots\cup\sigma(|T_{n}|)$. Let
$S_{j}=U_{j}f(|T_{j}|)\ (j=1,...,n)$ and ${\bf S}=(S_{1},...,S_{n})$. Let
$T_{1},...,T_{n}$ and $f$ satisfy $(1)$ and $(2)$. If
$(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})\in\sigma_{T}({\bf T}),\
\mbox{then }\
(e^{i\theta_{1}}f(r_{1}),...,e^{i\theta_{n}}f(r_{n}))\in\sigma_{T}({\bf S}).$

##  2 Proof of the theorem

First we need the following lemma.

######  Lemma  2.1.

Let ${\bf T}=(T_{1},...,T_{n})$ be a doubly commuting $n$-tuple of operators
and $T_{j}$ has property $(1)$ for $j=1,...,n$. Let $\\{D_{k}\\}$ be the chain
complex of $n$-tuple ${\bf T}=(T_{1},...,T_{n})$. If there exists some
$k\in\\{1,2,\cdots,n-1\\}$ and unit vectors
$x_{m}=\oplus_{j=1}^{r}x_{m}^{j}\in E_{k}^{n}(\mathcal{H})$ where
$r=\binom{n}{k}$, such that $(D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*})x_{m}\ \to\ 0$
as $m\to\infty$, then there exists $s\in\\{1,2,\cdots,r\\}$ such that
$\\{x_{m}^{s}\\}$ is a bounded below sequence of non-zero vectors of
$\mathcal{H}$ satisfying $T_{j}^{*}x_{m}^{s}\ \to\ 0$ as $m\to\infty$ for
$j=1,\cdots,n$. Thus, by taking unit vector
$y_{m}=\dfrac{x_{m}^{s}}{\|x_{m}^{s}\|}\in{\mathcal{H}}$, we have
$T_{j}^{*}y_{m}\ \to\ 0$ as $m\to\infty$ for $j=1,\cdots,n$.

###### Proof.

We show it by the mathematical induction.
(1) Let $n=2$. Then the chain complex of doubly commuting pair ${\bf
T}=(T_{1},T_{2})$ is

$0\ \stackrel{{\scriptstyle}}{{\ \longrightarrow\ }}\
\mathcal{H}\stackrel{{\scriptstyle D_{2}}}{{\ \longrightarrow\ }}\
\mathcal{H}\oplus\mathcal{H}\stackrel{{\scriptstyle D_{1}}}{{\
\longrightarrow\ }}\ \mathcal{H}\stackrel{{\scriptstyle}}{{\ \longrightarrow\
}}\ 0.$

By the definition of the Koszul complex we have

$D_{2}=\left(\begin{array}[]{cc}-T_{2}\\\ T_{1}\end{array}\right)\ \
\mbox{and}\ \ D_{1}=\left(\begin{array}[]{cc}T_{1}&T_{2}\end{array}\right).$

Since $T_{1},T_{2}$ are doubly commuting, we have

$\ \
D_{1}^{*}D_{1}+D_{2}D_{2}^{*}=\left(\begin{array}[]{cccc}T_{1}^{*}T_{1}+T_{2}T_{2}^{*}&0\\\
0&T_{1}T_{1}^{*}+T_{2}^{*}T_{2}\end{array}\right).$

Let $x_{m}=x_{m}^{1}\oplus x_{m}^{2}\in
E_{1}^{2}(\mathcal{H})\cong\mathcal{H}\oplus\mathcal{H}$ be unit vectors and

$\displaystyle(D_{1}^{*}D_{1}+D_{2}D_{2}^{*})x_{m}=\begin{pmatrix}T_{1}^{*}T_{1}+T_{2}T_{2}^{*}&0\\\
0&T_{1}T_{1}^{*}+T_{2}^{*}T_{2}\end{pmatrix}\begin{pmatrix}x_{m}^{1}\\\
x_{m}^{2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}(T_{1}^{*}T_{1}+T_{2}T_{2}^{*})x_{m}^{1}\\\
(T_{1}T_{1}^{*}+T_{2}^{*}T_{2})x_{m}^{2}\end{pmatrix}\ \to\ 0\ \text{as}\ \
m\to\infty.$

Since $\|x_{m}^{1}\|^{2}+\|x_{m}^{2}\|^{2}=1$ for all $m$, we may assume (i)
$x_{m}^{1}\not\rightarrow 0$ or (ii) $x_{m}^{2}\not\rightarrow 0$.

We assume (i). By taking subsequence, we may asume that there exists $0<c$
that that $0<c<\|x_{m}^{1}\|\leq 1$ for all $m$, i.e., bounded below. Then
$(T_{1}^{*}T_{1}+T_{2}T_{2}^{*})x_{m}^{1}\rightarrow 0$ implies
$T_{1}x_{m}^{1},T_{2}^{*}x_{m}^{1}\rightarrow 0$ and
$T_{1}^{*}x_{m}^{1}\rightarrow 0$ by (1). Case (ii) is similar. Hence the
statement holds for $n=2.$

(2) We assume that the statement holds for $(n-1)$-tuples of doubly commuting
operators. Asuume $(D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*})x_{m}\ \to\ 0$ as
$m\to\infty$ for unit vectors $x_{m}\in E_{k}^{n}(\mathcal{H})$.

Let $\\{F_{k}\\}$ be the chain complex of $(n-1)$-tuple ${\bf
T}^{\prime}=(T_{1},...,T_{n-1})$ and $x_{m}=y_{m}\oplus z_{m}\in
E_{k}^{n-1}(\mathcal{H})\oplus
E_{k-1}^{n-1}(\mathcal{H})=E_{k}^{n}(\mathcal{H})$. By Curto’s
characterization (see p.132, Curto [7]) it holds
$D_{k}=\left(\begin{array}[]{cc}F_{k}&(-1)^{k+1}{\rm diag}(T_{n})\\\
0&F_{k-1}\end{array}\right)$. Hence

$(D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*})x_{m}=\begin{pmatrix}\left(F_{k}^{*}F_{k}+F_{k+1}F_{k+1}^{*}+{\rm
diag}(T_{n}T_{n}^{*})\right)y_{m}\\\
\left(F_{k-1}^{*}F_{k-1}+F_{k}F_{k}^{*}+{\rm
diag}(T_{n}^{*}T_{n})\right)z_{m}\end{pmatrix}\rightarrow 0.$

Since $\|y_{m}\|^{2}+\|z_{m}\|^{2}=1$ for all $m$, we may assume (i)
$y_{m}\not\rightarrow 0$ or (ii) $z_{m}\not\rightarrow 0$.

We assume (i).

Then $\left(F_{k}^{*}F_{k}+F_{k+1}F_{k+1}^{*}+{\rm
diag}(T_{n}T_{n}^{*})\right)y_{m}\rightarrow 0$ implies
$\left(F_{k}^{*}F_{k}+F_{k+1}F_{k+1}^{*}\right)y_{m}\rightarrow 0$ and
$\left({\rm diag}(T_{n}T_{n}^{*})\right)y_{m}\rightarrow 0$. By taking
subsequence, we may asume that there exists $0<c$ that that $0<c<\|y_{m}\|\leq
1$ for all $m$. Let $v_{m}=\dfrac{y_{m}}{\|y_{m}\|}$. Then $v_{m}$ are unit
vectors and $\left(F_{k}^{*}F_{k}+F_{k+1}F_{k+1}^{*}\right)v_{m}\rightarrow 0$
and $\left({\rm diag}(T_{n}T_{n}^{*})\right)v_{m}\rightarrow 0$. Let
$v_{m}=\oplus_{s=1}^{\binom{n-1}{k}}v_{m}^{s}\in E^{n-1}_{k}({\mathcal{H}})$.
Then there exist $s\in\left\\{1,2,\cdots,\binom{n-1}{k}\right\\}$ such that
$v_{m}^{s}\in{\mathcal{H}}$ is a bounded below sequence of non-zero vectors
and $T_{j}^{*}v_{m}^{s}\rightarrow 0$ for $j=1,2,\cdots,n-1$ and
$T_{n}^{*}v_{m}^{s}\rightarrow 0$ as $m\rightarrow\infty$.

Case (ii) is similar. Hence the statement holds for $n.$ It completes the
proof. ∎

######  Theorem 2.2.

Let ${\bf T}=(T_{1},...,T_{n})$ be a doubly commuting $n$-tuple of operators
which satisfy that every $T_{j}\ (j=1,...,n)$ has property $(1)$. If
$z=(z_{1},...,z_{n})\in\sigma_{T}({\bf T})$, then there exists unit vectors
$y_{m}\in{\mathcal{H}}$ such that $(T_{j}-z_{j})^{*}y_{m}\ \to\ 0$ as
$m\to\infty$, that is,
$\overline{z}=(\overline{z_{1}},...,\overline{z_{n}})\in\sigma_{ja}({\bf
T}^{*})$, where ${\bf T}^{*}=(T_{1}^{*},...,T_{n}^{*})$.

###### Proof.

Since $z=(z_{1},...,z_{n})\in\sigma_{T}({\bf T})$, by the spectral mapping
theorem of the Taylor spectrum, it holds

$0=(0,...,0)\in\sigma_{T}({\bf T}-z),$

where ${\bf T}-z=(T_{1}-z_{1},...,T_{n}-z_{n})$. Since ${\bf T}-z$ is a doubly
commuting $n$-tuple of operators which satisfy that every $T_{j}-z_{j}\
(j=1,...,n)$ has property (1) and the Koszul complex $E({\bf T}-z)$ of
$n$-tuple ${\bf T}-z=(T_{1}-z_{1},...,T_{n}-z_{n})$ is not exact. Hence there
exists $k$ such that $(D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*})$ is not invertible.
Since the operator $D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*}$ is positive on the
space $E_{k}^{n}(\mathcal{H})$, there exists a sequence $\\{x_{m}\\}$ of unit
vectors of $E_{k}^{n}(\mathcal{H})$ such that
$\big{(}D_{k}^{*}D_{k}+D_{k+1}D_{k+1}^{*}\big{)}x_{m}\ \to\ 0$ as
$m\to\infty$. Hence, by Lemma 2.1 there exists a sequence $\\{y_{m}\\}$ of
unit vectors of $\mathcal{H}$ such that

$(T_{j}-z_{j})^{*}y_{m}\ \to\ 0\ \ \mbox{as}\ m\ \to\ \infty\ \ \mbox{for
all}\ \ j=1,...,n.$

It’s completes the proof. ∎

Proof of Theorem 1.9.
(1) If $n=2$, theorem holds by Theorem 2.3 of [6].
(2) We assume that the statment holds for $(n-1)$-tuple. Since
$(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})\in\sigma_{T}({\bf T})$, by
Theorem 2.2 there exists a sequence $\\{x_{m}\\}$ of unit vectors of
$\mathcal{H}$ such that $(T_{j}-r_{j}e^{i\theta_{j}})^{*}x_{m}\ \to\ 0$ as
$m\to\infty$ for all $j=1,...,n$. Consider the Berberian extension
${\mathcal{K}}$ of $\mathcal{H}$. Then there exists
$0\not=x^{\circ}\in{\mathcal{K}}$ such that

$(T_{j}^{\circ}-r_{j}e^{i\theta_{j}})^{*}x^{\circ}=0\ \ \mbox{for all}\ \
j=1,...,n.$

Let ${\mathcal{M}}=\ker(T_{n}^{\circ}-r_{n}e^{i\theta_{n}})^{*}.$ Then
${\mathcal{M}}(\not=\\{0\\})$ is a reducing subspace for
$T_{1}^{\circ},...,T_{n-1}^{\circ}$ and
$(r_{1}e^{i\theta_{1}},...,r_{n-1}e^{i\theta_{n-1}})\in\sigma_{T}({\bf
T}_{|{\mathcal{M}}}^{\circ^{\prime}})$, where ${\bf
T}_{|{\mathcal{M}}}^{\circ^{\prime}}=(T_{1|{\mathcal{M}}}^{\circ},...,T_{n-1|{\mathcal{M}}}^{\circ})$.
By the induction there exists a non-zero vector $y^{\circ}\in{\mathcal{M}}$
such that

$(S_{j}^{\circ}-e^{i\theta_{j}}f(r_{j}))^{*}y^{\circ}=0\ \ \mbox{for all}\ \
j=1,...,n-1.$

Let
${\mathcal{N}}=\displaystyle\bigcap_{j=1}^{n-1}\ker(S_{j}^{\circ}-e^{i\theta_{j}}f(r_{j}))^{*}.$
Then ${\mathcal{N}}$ is a reducing subspace for $T_{n}^{\circ}$. Let
${\mathcal{R}}={\mathcal{M}}\bigcap{\mathcal{N}}\not=\\{0\\}$. Hence
$r_{n}e^{i\theta_{n}}\in\sigma(T_{n|{\mathcal{R}}}^{\circ})$. By property (2)
there exists a non-zero vector $z^{\circ}\in{\mathcal{R}}$ such that
$(S_{n|{\mathcal{R}}}^{\circ}-e^{i\theta_{n}}f(r_{n}))^{*}z^{\circ}=0$. Since
this $z^{\circ}$ satisfies
$(S_{j|{\mathcal{R}}}^{\circ}-e^{i\theta_{j}}f(r_{j}))^{*}z^{\circ}=0$ for all
$j=1,...,n-1$, we have
$(e^{i\theta_{1}}f(r_{1}),...,e^{i\theta_{n}}f(r_{n}))\in\sigma_{T}({\bf S})$.
This completes the proof.

$\Box$

######  Corollary  2.3.

Let ${\bf T}=(T_{1},...,T_{n})$ be a doubly commuting $n$-tuple of
$p$-hyponormal operators $(0<p<1).$ Let $U_{j}$ be unitary for the polar
decomposition of $T_{j}=U_{j}|T_{j}|\ (j=1,...,n)$ and ${\bf
S}=\left(U_{1}|T_{1}|^{2p},...,U_{n}|T_{n}|^{2p}\right)$. Then

$\sigma_{T}({\bf S})\ =\
\\{(r_{1}^{2p}e^{i\theta_{1}},...,r_{n}^{2p}e^{i\theta_{n}})\,:\,(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})\in\sigma_{T}({\bf
T})\,\\}.$

###### Proof.

Let $f(t)=t^{2p}$ on the non-negative real line. Since ${\bf T}$ is a doubly
commuting $n$-tuple of $p$-hyponormal operators and $f(t)=t^{2p}$,
$T_{1},...,T_{n}$ and $f$ satisfy (2) and (3). Hence, by Theorem 1.9 we have

$\sigma_{T}({\bf S})\ \supset\
\\{(r_{1}^{2p}e^{i\theta_{1}},...,r_{n}^{2p}e^{i\theta_{n}})\,:\,(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})\in\sigma_{T}({\bf
T})\,\\}.$

Conversely, put $g(t)=t^{\frac{1}{2p}}$ on the non-negative real line. Since
${\bf S}$ is a doubly commuting pair of semi-hyponormal operators,
$S_{1},S_{2}$ and $g$ satisfy (2) and (3). Then we have the converse inclusion
by Theorem 1.9 and similar argument. ∎

######  Corollary  2.4.

Let ${\bf T}=(T_{1},...,T_{n})$ be a doubly commuting $n$-tuple of
$\log$-hyponormal operators with $\log|T_{j}|>0$. Let $U_{j}$ be unitary for
the polar decomposition of $T_{j}=U_{j}|T_{j}|\ (j=1,...,n)$ and ${\bf
S}=\left(U_{1}\log|T_{1}|,...,U_{n}\log|T_{n}|\right)$. Then

$\sigma_{T}({\bf S})=\\{e^{i\theta_{1}}\log r_{1},...,e^{i\theta_{n}}\log
r_{n})\,:\,(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})\in\sigma_{T}({\bf
T})\,\\}.$

###### Proof.

Let $f(t)=\log t$ on $(0,\infty)$. Since ${\bf T}$ is a doubly commuting
$n$-tuple of $\log$-hyponormal operators and $f(t)=\log t$, $T_{1},...,T_{n}$
and $f$ satisfy $(2)$ and $(3)$. So by Theorem 1.9 we have

$\sigma_{T}({\bf S})\ \supset\ \\{e^{i\theta_{1}}\log
r_{1},...,e^{i\theta_{n}}\log
r_{n})\,:\,(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})\in\sigma_{T}({\bf
T})\,\\}.$

Conversely, let $g(t)=e^{t}$ on the non-negative real line. Since ${\bf S}$ is
a doubly commuting $n$-tuple of semi-hyponormal operators, $S_{1},...,S_{n}$
and $g$ satisfy (2) and (3). Hence, we have the converse inclusion by similar
argument. ∎

Acknowledgment. This research is partially supported by RIMS.

## References

  * [1] S.K. Berberian, Approximate proper vectors, Proc. Amer. Math. Soc. 13 (1962), 111-114.
  * [2] M. Chō, Relation between the Taylor spectrum and the Xia spectrum, Proc. Amer. Math. Soc., 128 (1999), 2357-2363.
  * [3] M. Chō and T. Huruya, $p$-Hyponormal operators for $0<p<\frac{1}{\,2\,}$, Comment. Math. 33 (1993), 23-29.
  * [4] M. Chō and M. Itoh, Putnam’s inequality for $p$-hyponormal operators, Proc. Amer. Math. Soc 123 (1995), 2435-2440.
  * [5] M. Chō, H. Motoyoshi and B. Načevska Nastovska, On the joint spectra of commuting tuples of operators and a conjugation, Functional Analysis, Approximation and Computation, 9:2 (2017), 21-26.
  * [6] M. Chō and K. Tanahashi, New spectral mapping theorem of the Taylor spectrum, Sci. Math. Japon, 84 No.2 (2021), 145-154.
  * [7] R. Curto, Fredholm and invertible $n$-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), 129-159.
  * [8] K. Tanahashi, Putnam’s inequality for $\log$-hyponormal operators, Integr. Equat. Oper. Th. 48(2004), 103-114.
  * [9] J.L.Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970), 172-191.
  * [10] J.L.Taylor, The analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1-38.
  * [11] F.-H. Vasilescu, On pairs of commuting operators, Studia Math. 62 (1978), 203-207.
  * [12] D. Xia, On the semi-hyponormal $n$-tuple of operators, Integr. Equat. Oper. Th. 6(1983), 879-898.
  * [13] D. Xia, Spectral Theory of Hyponormal operators Birkh$\ddot{{\rm a}}$user 1983.

Muneo Chō

15-3-1113, Tsutsui-machi Yahatanishi-ku, Kita-kyushu 806-0032, Japan

e-mail<EMAIL_ADDRESS>
B. Načevska Nastovska

Faculty of Electrical Engineering and Information Technologies, ”Ss Cyril and
Methodius“ University in Skopje, Macedonia

e-mail<EMAIL_ADDRESS>
Kôtarô Tanahashi

Department of Mathematics, Tohoku Medical and Pharmaceutical University,
Sendai 981-8558, Japan

e-mail<EMAIL_ADDRESS>