# Doubly commuting and dual doubly commuting semigroups of isometries

Tirthankar Bhattacharyya, Shubham Rastogi and Vijaya Kumar U  Department of
Mathematics
Indian Institute of Science
Bangalore 560012, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Structures of commuting semigroups of isometries under certain additional
assumptions like double commutativity or dual double commutativity are found.

**footnotetext: MSC: Primary: 47D03, 47A65. Secondary: 47B91.
Keywords: The shift semigroup, Isometric semigroups, Doubly commuting, Dual
doubly commuting.

##  1\. Introduction

One of the building blocks of $C_{0}$-semigroups of isometries is the right-
shift-semigroup
$\mathscr{S}^{\mathcal{F}}=(\mathscr{S}^{\mathcal{F}}_{t})_{t\geq 0}$ on
$L^{2}(\mathbb{R}^{+},\mathcal{F})$ for any Hilbert space $\mathcal{F}.$ It is
defined as

$(\mathscr{S}_{t}^{\mathcal{F}}f)x=\begin{cases}f(x-t)&\text{if }x\geq t,\\\
0&\text{otherwise,}\end{cases}$

for $f\in L^{2}(\mathbb{R}_{+},\mathcal{F})$ and it has been shown by Cooper
in [5] to be one of the direct summands in the structure theorem for any
$C_{0}$-semigroup of isometries. We shall denote
$\mathscr{S}_{t}^{\mathbb{C}}$ by just $\mathscr{S}_{t}.$ Sometimes we shall
use the identification of $L^{2}(\mathbb{R}_{+},\mathcal{F})$ with
$L^{2}(\mathbb{R}_{+})\otimes\mathcal{F}$ which transforms
$\mathscr{S}_{t}^{\mathcal{F}}$ into $\mathscr{S}_{t}\otimes I_{\mathcal{F}}.$

All semigroups in this note are $C_{0}$-semigroups (i.e., strongly continuous
one parameter semigroups). Two such semigroups $V_{1}=(V_{1,t})_{t\geq 0}$ and
$V_{2}=(V_{2,s})_{s\geq 0}$ on a Hilbert space $\mathcal{H}$ are said to be

  1. (1)

commuting if

$V_{1,t}V_{2,s}=V_{2,s}V_{1,t}\text{ for all }t,s\geq 0.$

  2. (2)

doubly commuting if they are commuting and

$V_{1,t}V_{2,s}^{*}=V_{2,s}^{*}V_{1,t}\text{ for all }t,s\geq 0.$

We completely characterize pairs of doubly commuting semigroups of isometries
in terms of concrete models in 3.5 and Corollary 3.7. See [2] and [3] for a
preliminary result of Cooper type decomposition obtained by Bînzar and
Lăzureanu.

This can be done in two ways. To begin with, we use the identification of the
right-shift-semigroup with a semigroup of certain multiplication operators on
a vector valued Hardy space given in [1, Theorem 3.4]. This gives rise to
certain partial isometries. We find commutants of these partial isometries.
This gives a proof of 3.5. On the other hand, the result [1, Lemma 12.1] of
Berger-Coburn-Lebow is also useful for proving 3.5.

The minimal unitary extension of a pair of commuting semigroups of isometries
is known due to [14, 11]. Therefore, the dual can be defined for a pair of
commuting semigroups of isometries. We obtain a Cooper type decomposition for
a pair of dual doubly commuting semigroups of isometries in 4.6 and a model in
4.7.

We state below Cooper’s theorem. The motivation comes from the celebrated Wold
decomposition in [16]. A semigroup $V=(V_{t})_{t\geq 0}$ of isometries on a
Hilbert space $\mathcal{H}$ is said to be completely nonunitary (c.n.u) if
there is no reducing subspace $\mathcal{H}_{0}$ of $\mathcal{H}$ such that
$V|_{\mathcal{H}_{0}}=(V_{t}|_{\mathcal{H}_{0}})_{t\geq 0}$ is a unitary
semigroup.

######  Theorem 1.1 (Cooper [5]).

Let $V=(V_{t})_{t\geq 0}$ be a strongly continuous semigroup of isometries on
a Hilbert space $\mathcal{H}.$ Then $\mathcal{H}$ uniquely decomposes as the
direct sum

$\mathcal{H}=\mathcal{H}_{p}\oplus\mathcal{H}_{u}$

of reducing subspaces $\mathcal{H}_{p}$ and $\mathcal{H}_{u}$ of $V,$ so that

  1. (1)

$V|_{\mathcal{H}_{p}}$ is a c.n.u semigroup of isometries, and

  2. (2)

$V|_{\mathcal{H}_{u}}$ is a unitary semigroup.

Moreover, any c.n.u semigroup of isometries is unitarily equivalent to the
right-shift-semigroup
$\mathscr{S}^{\mathcal{F}}=(\mathscr{S}_{t}^{\mathcal{F}})_{t\geq 0}$ on
$L^{2}(\mathbb{R}_{+},\mathcal{F})$ for some Hilbert space $\mathcal{F}.$

An astute reader would notice that this paper is motivated by structure
theorems for commuting isometries obtained over decades in [4], [8], [9] and
[10] to quote a few.

##  2\. On the right-shift-semigroup

For any Hilbert space $\mathcal{F},$ let $H^{2}({\mathbb{D}},\mathcal{F})$
denote the Hardy space of $\mathcal{F}$-valued analytic functions over the
unit disc ${\mathbb{D}}.$ We just write $H^{2}({\mathbb{D}})$ instead of
$H^{2}({\mathbb{D}},\mathbb{C}).$ The space $H^{2}({\mathbb{D}},\mathcal{F})$
is naturally identified with $H^{2}({\mathbb{D}})\otimes\mathcal{F}.$

It is well known that (see [15]) the right-shift-semigroup
$\mathscr{S}^{\mathcal{F}}$ on $L^{2}(\mathbb{R}_{+},\mathcal{F})$ is
unitarily equivalent to the semigroup $(M_{\psi_{t}})_{t\geq 0}$ of
multiplication operators on the vector valued Hardy space
$H^{2}({\mathbb{D}},\mathcal{F}),$ where
$\psi_{t}:{\mathbb{D}}\to\mathscr{B}(\mathcal{F})$ are the operator valued
bounded analytic functions defined by

$\psi_{t}(z)=e^{t\frac{z+1}{z-1}}I_{\mathcal{F}}$

for $z\in{\mathbb{D}}$ and $t\geq 0.$

On the other hand, in [1, Theorem 3.4], Berger, Coburn and Lebow obtained
another identification of the right-shift-semigroup
$\mathscr{S}^{\mathcal{F}}$ with a semigroup of multiplication operators on
the vector valued Hardy space $H^{2}({\mathbb{D}})\otimes
L^{2}(\mathbb{T},\mathcal{F})$ where $\mathbb{T}$ is the unit circle and
$\mathcal{F}$ is a separable Hilbert space. In this section, we recall their
identification of $\mathscr{S}^{\mathcal{F}}$ rephrased on the vector valued
Hardy space $H^{2}({\mathbb{D}},L^{2}([0,1],\mathcal{F})).$ The subsequent
lemma is of independent interest and is also useful to give different proofs
of 3.5 and 3.6 for the case of separable Hilbert spaces.

######  Theorem 2.1.

The right-shift-semigroup
$\mathscr{S}^{\mathcal{F}}=(\mathscr{S}_{t}^{\mathcal{F}})_{t\geq 0}$ on
$L^{2}(\mathbb{R}_{+},\mathcal{F})$ is unitarily equivalent to the semigroup
$(M_{\varphi_{t}^{\mathcal{F}}})_{t\geq 0}$ on
$H^{2}({\mathbb{D}},L^{2}([0,1],\mathcal{F})),$ where for $n\in\mathbb{Z}_{+}$
and $n\leq t<n+1,$ the multiplier
$\varphi^{\mathcal{F}}_{t}:{\mathbb{D}}\to\mathscr{B}(L^{2}([0,1],\mathcal{F}))$
is given by

$\varphi^{\mathcal{F}}_{t}(z)=E^{\mathcal{F}}_{0,t-n}z^{n}+E^{\mathcal{F}}_{1,t-n}z^{n+1}$
(2.1)

and for $0\leq s<1,$ $E^{\mathcal{F}}_{0,s},E^{\mathcal{F}}_{1,s}$ are the
partial isometries in $\mathscr{B}(L^{2}([0,1],\mathcal{F}))$ given by

$(E^{\mathcal{F}}_{0,s}f)(x)=\begin{cases}0&\text{if }x<s,\\\ f(x-s)&\text{if
}s\leq x\leq 1,\end{cases}$ (2.2)

and

$(E^{\mathcal{F}}_{1,s}f)(x)=\begin{cases}f(1-s+x)&\text{if }x\leq s,\\\
0&\text{if }s<x\leq 1.\end{cases}$ (2.3)

###### Proof.

Consider the unitary $W:L^{2}(\mathbb{R}_{+},\mathcal{F})\to
H^{2}({\mathbb{D}},L^{2}([0,1],\mathcal{F}))$ defined by

$Wf=\sum_{n=0}^{\infty}f_{n}z^{n}$ (2.4)

where $f_{n}(\alpha):=f(n+\alpha)$ for $\alpha\in[0,1]$ and
$n\in\mathbb{Z}_{+}.$ Let $0\leq t<1$ and $g\in L^{2}([0,1],\mathcal{F}).$

$W\mathscr{S}_{t}^{\mathcal{F}}W^{*}(gz^{n})=W\mathscr{S}_{t}^{\mathcal{F}}(h)=(E^{\mathcal{F}}_{0,t}g)z^{n}+(E^{\mathcal{F}}_{1,t}g)z^{n+1}=M_{\varphi^{\mathcal{F}}_{t}}(gz^{n}),$

where $h\in L^{2}(\mathbb{R}_{+},\mathcal{F})$ is given by

$h(\alpha)=\begin{cases}g(\alpha-n)&\text{if }n\leq\alpha\leq n+1,\\\
0&\text{else.}\end{cases}$

It is evident that
$W\mathscr{S}_{1}^{\mathcal{F}}W^{*}=M_{\varphi_{1}^{\mathcal{F}}}=M_{z}^{L^{2}([0,1],\mathcal{F})},$
the multiplication operator by $z$ on
$H^{2}({\mathbb{D}},L^{2}([0,1],\mathcal{F})).$ Hence
$W\mathscr{S}_{t}^{\mathcal{F}}W^{*}=M_{\varphi^{\mathcal{F}}_{t}}$ for $0\leq
t\leq 1.$ For $n\in\mathbb{N}$ and $n\leq t<n+1,$ since
$\mathscr{S}_{t}^{\mathcal{F}}=(\mathscr{S}_{1}^{\mathcal{F}})^{n}\mathscr{S}_{t-n}^{\mathcal{F}},$
we get

$\displaystyle W\mathscr{S}_{t}^{\mathcal{F}}W^{*}$
$\displaystyle=W(\mathscr{S}_{1}^{\mathcal{F}})^{n}W^{*}W\mathscr{S}_{t-n}^{\mathcal{F}}W^{*}$
$\displaystyle=(W\mathscr{S}_{1}^{\mathcal{F}}W^{*})^{n}W\mathscr{S}_{t-n}^{\mathcal{F}}W^{*}$
$\displaystyle=(M_{z}^{L^{2}([0,1],\mathcal{F})})^{n}M_{\varphi^{\mathcal{F}}_{t-n}}$
$\displaystyle=M_{z^{n}\varphi^{\mathcal{F}}_{t-n}}$
$\displaystyle=M_{\varphi^{\mathcal{F}}_{t}}.$

This completes the proof. ∎

######  Lemma 2.2.

Let $\mathcal{F}$ be a separable Hilbert space and
$\Lambda:L^{2}[0,1]\otimes\mathcal{F}\to L^{2}([0,1],\mathcal{F})$ be the
natural unitary isomorphism. Suppose
$B\in\mathscr{B}(L^{2}([0,1],\mathcal{F}))$ commutes with
$E^{\mathcal{F}}_{0,s}$ and $E^{\mathcal{F}}_{1,s}$ for all $0\leq s<1.$ Then,
$B=\Lambda(I_{L^{2}[0,1]}\otimes C)\Lambda^{*}$ for some
$C\in\mathscr{B}(\mathcal{F}).$

###### Proof.

For any $0\leq s<1,$ note that

$\operatorname{ran}E^{\mathcal{F}}_{0,s}=\\{f:\operatorname{supp}(f)\subseteq[s,1]\\}\
and\
\operatorname{ran}E^{\mathcal{F}}_{1,s}=\\{f:\operatorname{supp}(f)\subseteq[0,s]\\}.$

Hence
$L^{2}([0,1],\mathcal{F})=\operatorname{ran}E^{\mathcal{F}}_{0,s}\oplus\operatorname{ran}E^{\mathcal{F}}_{1,s}$
for any $0\leq s<1.$ Since $B$ commutes with $E^{\mathcal{F}}_{0,s}$ and
$E^{\mathcal{F}}_{1,s}$ for all $0\leq s<1,$ the ranges of
$E^{\mathcal{F}}_{0,s}$ and the ranges of $E^{\mathcal{F}}_{1,s}$ are reduced
by $B.$ For any $x\in\mathcal{F}$ and an interval $I\subset[0,1],$ let
$\mathbf{1}_{I}\cdot x:[0,1]\to\mathcal{F}$ denotes the function

$(\mathbf{1}_{I}\cdot x)(\alpha)=\begin{cases}x&\text{if }\alpha\in I,\\\
0&\text{else.}\end{cases}$

Let $f_{x}:=B(\mathbf{1}_{[0,1]}\cdot x).$ Note that

$B(\mathbf{1}_{[0,1]}\cdot x)=B(\mathbf{1}_{[0,c]}\cdot
x)+B(\mathbf{1}_{(c,d)}\cdot x)+B(\mathbf{1}_{[d,1]}\cdot x)=f_{x}$

As $B$ reduces the ranges of $E^{\mathcal{F}}_{1,c}$ and
$E^{\mathcal{F}}_{0,d},$ we have

$B(\mathbf{1}_{(c,d)}\cdot x)(\alpha)=\begin{cases}f_{x}(\alpha)&\text{if
}\alpha\in(c,d),\\\ 0&\text{else.}\end{cases}$

Hence $B(\mathbf{1}_{(c,d)}\cdot x)=\mathbf{1}_{(c,d)}f_{x}$ for all $0\leq
c<d\leq 1.$ With a little more work, $f_{x}$ can be shown to be a bounded
measurable function (and not just in $L^{2}([0,1],\mathcal{F})$), but we do
not need this fact.

Since $B$ commutes with $E^{\mathcal{F}}_{0,s}$ and $E^{\mathcal{F}}_{1,s}$
for $0\leq s<1,$ we have

$B(\mathbf{1}_{[s,1]}\cdot x)=E^{\mathcal{F}}_{0,s}(f_{x})\quad\text{and}\quad
B(\mathbf{1}_{[0,s]}\cdot x)=E^{\mathcal{F}}_{1,s}(f_{x}).$

Therefore,

$B(\mathbf{1}_{[0,1]}\cdot
x)=E^{\mathcal{F}}_{0,s}(f_{x})+E^{\mathcal{F}}_{1,s}(f_{x})=f_{x}\text{ for
}0\leq s<1.$ (2.5)

Extend $f_{x}$ on $\mathbb{R}$ periodically with period $1,$ say
$\tilde{f_{x}}.$ Note that the translation of $\tilde{f_{x}}$ by any $0\leq
s<1$ equals $\tilde{f_{x}}$ almost everywhere by Eq. 2.5. This implies that
$f_{x}=\mathbf{1}_{[0,1]}\cdot y$ for some $y\in\mathcal{F}.$ See also [12,
Problem 4, Chap. 7].

Let $\Theta:\mathcal{F}\to L^{2}([0,1],\mathcal{F})$ be the isometric
embedding $\Theta x=\mathbf{1}_{[0,1]}\cdot x.$ Use $\Theta$ to define $C$ by
$C=\Theta^{*}B\Theta.$ Let $\Lambda:L^{2}[0,1]\otimes\mathcal{F}\to
L^{2}([0,1],\mathcal{F})$ be the natural unitary. Then

$\displaystyle B(\mathbf{1}_{(c,d)}\cdot x)$
$\displaystyle=\mathbf{1}_{(c,d)}B(\Theta(x))$
$\displaystyle=\mathbf{1}_{(c,d)}\cdot\Theta^{*}B\Theta(x)$
$\displaystyle=\mathbf{1}_{(c,d)}\cdot Cx$
$\displaystyle=\Lambda(I_{L^{2}[0,1]}\otimes
C)\Lambda^{*}(\mathbf{1}_{(c,d)}\cdot x).$

Since the characteristic functions $\mathbf{1}_{(c,d)}$ for $0\leq c<d\leq 1$
are total in $L^{2}[0,1],$ we get $B=\Lambda(I_{L^{2}[0,1]}\otimes
C)\Lambda^{*}.$ This completes the proof. ∎

##  3\. Doubly commuting semigroups of isometries

######  Example 3.1.

For $j=1,2$ define $\mathcal{S}_{j,t}:L^{2}(\mathbb{R}_{+}^{2})\to
L^{2}(\mathbb{R}_{+}^{2})$ by

$\displaystyle(\mathcal{S}_{1,t}f)(x_{1},x_{2})=\begin{cases}f(x_{1}-t,x_{2})&\text{if
}x_{1}\geq t\\\
0&\text{else}\end{cases},(\mathcal{S}_{2,t}f)(x_{1},x_{2})=\begin{cases}f(x_{1},x_{2}-t)&\text{if
}x_{2}\geq t\\\ 0&\text{else.}\end{cases}$

Then $\mathcal{S}_{1}:=(\mathcal{S}_{1,t})_{t\geq 0}$ and
$\mathcal{S}_{2}:=(\mathcal{S}_{2,t})_{t\geq 0}$ are doubly commuting
semigroups of isometries on $L^{2}(\mathbb{R}_{+}^{2}).$

A pair of semigroups $(V_{1},V_{2})$ on $\mathcal{H}$ is said to be jointly
unitarily equivalent to $(W_{1},W_{2})$ on $\mathcal{K}$ if there is a unitary
$U:\mathcal{H}\to\mathcal{K}$ such that

$W_{j,t}=UV_{j,t}U^{*}\text{ for all }t\geq 0,j=1,2.$

In the natural isomorphism of $L^{2}(\mathbb{R}_{+}^{2})$ and
$L^{2}(\mathbb{R}_{+})\otimes L^{2}(\mathbb{R}_{+})$, we can see that the pair
of semigroups of isometries $(\mathcal{S}_{1},\mathcal{S}_{2})$ is jointly
unitarily equivalent to the pair of semigroups of isometries
$(\mathscr{S}\otimes
I_{L^{2}(\mathbb{R}_{+})},I_{L^{2}(\mathbb{R}_{+})}\otimes\mathscr{S}),$ where
$\mathscr{S}$ is the right-shift-semigroup on $L^{2}(\mathbb{R}_{+}).$

######  Definition 3.2.

A pair $(V_{1},V_{2})$ of semigroups is called a bishift-semigroup if
$(V_{1},V_{2})$ is jointly unitarily equivalent to $(\mathcal{S}_{1}\otimes
I_{\mathcal{F}},\mathcal{S}_{2}\otimes I_{\mathcal{F}})$ on
$L^{2}(\mathbb{R}_{+}^{2})\otimes\mathcal{F}$ for some Hilbert space
$\mathcal{F}.$

The following theorem is known due to Bînzar and Lăzureanu [2, 3].

######  Theorem 3.3 ([3]).

Let $V_{1}$ and $V_{2}$ be doubly commuting semigroups of isometries on
$\mathcal{H},$ then there exists a decomposition

$\mathcal{H}=\mathcal{H}_{p,p}\oplus\mathcal{H}_{p,u}\oplus\mathcal{H}_{u,p}\oplus\mathcal{H}_{u,u}$

where $\mathcal{H}_{i,j}$ reduces both $V_{1}$ and $V_{2}$ for all
$i,j\in\\{p,u\\}$ such that
(1) both $V_{1}$ and $V_{2}$ are c.n.u semigroups on $\mathcal{H}_{p,p},$
(2) $V_{1}$ is a c.n.u semigroup, $V_{2}$ is a unitary semigroup on
$\mathcal{H}_{p,u},$
(3) $V_{1}$ is a unitary semigroup, $V_{2}$ is a c.n.u semigroup on
$\mathcal{H}_{u,p}$ and
(4) both $V_{1}$ and $V_{2}$ are unitary semigroups on $\mathcal{H}_{u,u}.$

In the rest of this section we shall give models for
$\mathcal{H}_{p,p},\mathcal{H}_{p,u}$ and $\mathcal{H}_{u,p}$ parts in the
theorem above. The following lemma describes the operators doubly commuting
with the shift. Let $M_{z}$ denote the multiplication by $z$ on the Hardy
space of the unit disc $H^{2}({\mathbb{D}}).$

######  Lemma 3.4.

If $N$ is a bounded operator on $H^{2}({\mathbb{D}})\otimes\mathcal{F}$ doubly
commuting with $M_{z}\otimes I_{\mathcal{F}},$ then
$N=I_{H^{2}({\mathbb{D}})}\otimes\omega$ for some bounded operator $\omega$ on
$\mathcal{F}.$ In particular, any normal operator $N$ commuting with
$M_{z}\otimes I_{\mathcal{F}},$ is of the form
$N=I_{H^{2}({\mathbb{D}})}\otimes\omega$ for some normal operator $\omega$ on
$\mathcal{F}.$

###### Proof.

A proof of the first part of the lemma can be found in [6, page 38]. The part
involving normality is then straightforward from Fuglede’s theorem ([7]). ∎

######  Theorem 3.5.

Let $V_{1}$ and $V_{2}$ be two semigroups on $\mathcal{H}$. Then $V_{1}$ and
$V_{2}$ are doubly commuting c.n.u semigroups of isometries if and only if
$(V_{1},V_{2})$ is a bishift-semigroup.

###### Proof.

Let $\mathcal{A}=C^{*}\\{V_{2,s}:s\geq 0\\}\subseteq\mathscr{B}(\mathcal{H})$
be the $C^{*}$-algebra generated by $\\{V_{2,s}:s\geq 0\\}.$ Suppose $V_{1}$
and $V_{2}$ are doubly commuting, every element of $\mathcal{A}$ commutes with
$V_{1,t}$ for all $t\geq 0.$ Therefore, by [1, Lemma 12.1] there is a Hilbert
space $\mathcal{G}$ and a unitary $U:\mathcal{H}\to
L^{2}(\mathbb{R}_{+})\otimes\mathcal{G}$ such that

$UV_{1,t}U^{*}=\mathscr{S}_{t}\otimes I_{\mathcal{G}}\text{ and }\quad
UV_{2,t}U^{*}=I_{L^{2}(\mathbb{R}_{+})}\otimes B_{t}$

for all $t\geq 0,$ where $B_{t}\in\mathscr{B}(\mathcal{G}).$ As
$(V_{2,t})_{t\geq 0}$ is a c.n.u semigroup of isometries, we must have
$(B_{t})_{t\geq 0}$ to be a c.n.u semigroup of isometries on $\mathcal{G}.$
Therefore, by 1.1, there is a Hilbert space $\mathcal{F}$ and a unitary
$W:\mathcal{G}\to L^{2}(\mathbb{R}_{+})\otimes\mathcal{F}$ such that

$WB_{t}W^{*}=\mathscr{S}_{t}\otimes I_{\mathcal{F}}\text{ for }t\geq 0.$

Let $Z:\mathcal{H}\to L^{2}(\mathbb{R}_{+})\otimes
L^{2}(\mathbb{R}_{+})\otimes\mathcal{F}$ be the unitary $Z=(I\otimes W)U.$
Then

$\displaystyle ZV_{1,t}Z^{*}$ $\displaystyle=(I_{L^{2}(\mathbb{R}_{+})}\otimes
W)(\mathscr{S}_{t}\otimes I_{\mathcal{G}})(I_{L^{2}(\mathbb{R}_{+})}\otimes
W^{*})=\mathscr{S}_{t}\otimes I_{L^{2}(\mathbb{R}_{+})}\otimes
I_{\mathcal{F}}\text{ and }$ $\displaystyle ZV_{2,t}Z^{*}$
$\displaystyle=(I_{L^{2}(\mathbb{R}_{+})}\otimes
W)(I_{L^{2}(\mathbb{R}_{+})}\otimes B_{t})(I_{L^{2}(\mathbb{R}_{+})}\otimes
W^{*})=I_{L^{2}(\mathbb{R}_{+})}\otimes\mathscr{S}_{t}\otimes I_{\mathcal{F}}$

for all $t\geq 0.$ This shows that $(V_{1},V_{2})$ is a bishift-semigroup. ∎

We provide another proof of 3.5 using Lemma 2.2 and Lemma 3.4 when
$\mathcal{H}$ is separable.

###### Another proof of 3.5.

Since $V_{1}$ is a c.n.u semigroup, by 1.1 and 2.1, there exists a separable
Hilbert space $\mathcal{F}$ and a unitary $U:\mathcal{H}\to
H^{2}({\mathbb{D}},L^{2}([0,1],\mathcal{F}))$ such that
$UV_{1,t}U^{*}=M_{\varphi^{\mathcal{F}}_{t}}$ where
$\varphi^{\mathcal{F}}_{t}$ is as given in Eq. 2.1. Then $UV_{2,t}U^{*}$
doubly commutes with
$M_{\varphi^{\mathcal{F}}_{1}}=M_{z}^{L^{2}([0,1],\mathcal{F})}$ for all
$t\geq 0.$ Therefore, by Lemma 3.4, $UV_{2,t}U^{*}=M_{\psi_{t}}$ where
$\psi_{t}$ is the constant function given by $\psi_{t}(z)=B_{t}$ for all
$z\in{\mathbb{D}}.$ This shows that $(V_{1},V_{2})$ is jointly unitarily
equivalent to $((M_{\varphi^{\mathcal{F}}_{t}})_{t},(M_{\psi_{t}})_{t})$ on
$H^{2}({\mathbb{D}},L^{2}([0,1],\mathcal{F})).$

Note that $B=(B_{t})_{t\geq 0}$ is a c.n.u semigroup of isometries on
$L^{2}([0,1],\mathcal{F}).$ Since $\psi_{t}$ commutes with
$\varphi^{\mathcal{F}}_{s},$ $B_{t}$ commutes with $E^{\mathcal{F}}_{0,s}$ and
$E^{\mathcal{F}}_{1,s}$ for all $t\geq 0$ and $0\leq s<1.$ Therefore by Lemma
2.2, $\Lambda^{*}\psi_{t}(z)\Lambda=I_{L^{2}([0,1])}\otimes C_{t}$ for all
$z\in{\mathbb{D}},$ where $C=(C_{t})$ is a c.n.u semigroup of isometries on
$\mathcal{F}$ and $\Lambda:L^{2}[0,1]\otimes\mathcal{F}\to
L^{2}([0,1],\mathcal{F})$ is the natural unitary. Note also that
$\Lambda^{*}\varphi^{\mathcal{F}}_{t}(z)\Lambda=\varphi^{\mathbb{C}}_{t}(z)\otimes
I_{\mathcal{F}}$ for all $z\in{\mathbb{D}}$ and $t\geq 0,$ since
$\Lambda^{*}E^{\mathcal{F}}_{j,s}\Lambda=E^{\mathbb{C}}_{j,s}\otimes
I_{\mathcal{F}}$ for $j=0,1$ and for all $0\leq s<1.$

Therefore, $((M_{\varphi^{\mathcal{F}}_{t}})_{t},(M_{\psi_{t}})_{t})$ is
jointly unitarily equivalent to $(\mathscr{S}\otimes
I_{\mathcal{F}},I_{L^{2}(\mathbb{R}_{+})}\otimes C_{t})$ by 2.1, and this is
jointly unitarily equivalent to $(\mathscr{S}\otimes
I_{L^{2}(\mathbb{R}_{+})}\otimes
I_{\mathcal{K}},I_{L^{2}(\mathbb{R}_{+})}\otimes\mathscr{S}\otimes
I_{\mathcal{K}})$ for some Hilbert space $\mathcal{K},$ by 1.1, as $(C_{t})$
is a c.n.u semigroup of isometries. This completes the proof. ∎

We observe the following result which is analogous to Lemma 3.4. The proof
follows from [1, Lemma 12.1] and Fuglede’s theorem [7]. We remark that an
elementary proof can be given using the notion of cogenerators and Lemma 3.4
instead of invoking [1, Lemma 12.1]. We leave that to the reader.

######  Theorem 3.6.

Let $A=(A_{t})_{t\geq 0}$ be a semigroup of bounded operators on
$L^{2}(\mathbb{R}_{+})\otimes\mathcal{F}.$ Then, $A$ doubly commutes with
$\mathscr{S}\otimes I_{\mathcal{F}}$ if and only if there exists a semigroup
$(B_{t})_{t\geq 0}$ on $\mathcal{F}$ such that $A_{t}=I\otimes B_{t}$ for all
$t\geq 0.$ In particular, a normal semigroup $A$ commutes with
$\mathscr{S}\otimes I_{\mathcal{F}}$ if and only if there exists a normal
semigroup $(B_{t})_{t\geq 0}$ on $\mathcal{F}$ such that $A_{t}=I\otimes
B_{t}$ for all $t\geq 0.$

As a corollary to the above theorem we get the model for the
$\mathcal{H}_{p,u}$ (equivalently, for the $\mathcal{H}_{u,p}$) part.

######  Corollary 3.7.

Let $V_{1}$ be a c.n.u semigroup of isometries commuting with a unitary
semigroup $V_{2}$ on $\mathcal{H}.$ Then, there is a Hilbert space
$\mathcal{F}$ and a unitary isomorphism between $\mathcal{H}$ and
$L^{2}(\mathbb{R}_{+})\otimes\mathcal{F}$ so that under the unitary
isomorphism, $(V_{1},V_{2})$ is jointly equivalent to $(\mathscr{S}\otimes
I_{\mathcal{F}},I_{L^{2}(\mathbb{R}_{+})}\otimes U)$ for some unitary
semigroup $U$ on $\mathcal{F},$ where $\mathscr{S}$ is the right-shift-
semigroup on $L^{2}(\mathbb{R}_{+}).$

##  4\. Dual doubly commuting semigroups of isometries

Let $(V_{1},V_{2})$ be a pair of commuting semigroups of isometries on
$\mathcal{H},$ where $V_{j}=(V_{j,t})_{t\geq 0}$ for $j=1,2.$ Let $(\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu,\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern 1.5mu)$ on $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern 1.5mu$ be the
minimal unitary extension of the pair $(V_{1},V_{2}),$ (this extension exists
due to [14, 11]). So, $\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu=(\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu)_{t\in\mathbb{R}}$
and $\mkern 1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern 1.5mu=(\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern 1.5mu)_{s\in\mathbb{R}}$
are commuting unitary groups on $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern 1.5mu$ such that
$V_{1,t}V_{2,s}=P_{\mathcal{H}}\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern 1.5mu|_{\mathcal{H}}$
for $t,s\geq 0$ and

$\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu=\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu(\mathcal{H}):s,t\in\mathbb{R}\\}.$

It is clear that $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu\ominus\mathcal{H}$ is an invariant subspace for $(\mkern
1.5mu\overline{\mkern-1.5muV_{i,t}\mkern-1.5mu}\mkern 1.5mu)^{*}$ for all
$t\geq 0,i=1,2.$ Let $\widetilde{V_{i,t}}:={(\mkern
1.5mu\overline{\mkern-1.5muV_{i,t}\mkern-1.5mu}\mkern 1.5mu)}^{*}|_{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu\ominus\mathcal{H}}$ for $i=1,2,t\geq 0.$ Then
$(\widetilde{V_{1}},\widetilde{V_{2}})$ is a pair of commuting semigroups of
isometries on $\widetilde{\mathcal{H}}:=\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu\ominus\mathcal{H},$ where
$\widetilde{V_{1}}=(\widetilde{V_{1,t}})_{t\geq 0}$ and
$\widetilde{V_{2}}=(\widetilde{V_{2,s}})_{s\geq 0}.$ The pair
$(\widetilde{V_{1}},\widetilde{V_{2}})$ is called the dual of $(V_{1},V_{2}).$
The pair $(V_{1},V_{2})$ is said to be dual doubly commuting if the dual
$(\widetilde{V_{1}},\widetilde{V_{2}})$ is doubly commuting. In the sequel,
for a semigroup $W=(W_{t})_{t\geq 0}$ the notation $W^{*}$ denotes the adjoint
semigroup $(W_{t}^{*})_{t\geq 0}.$

######  Example 4.1.

Let $\mathcal{H}=L^{2}(\mathbb{R}^{2}\setminus\mathbb{R}^{2}_{+})$ and for
$f\in\mathcal{H},$ define

$\displaystyle(\mathcal{M}_{1,t}f)(x_{1},x_{2})$
$\displaystyle=\begin{cases}0&\text{if }x_{2}\geq 0\text{ and }x_{1}\geq-t,\\\
f(x_{1}+t,x_{2})&\text{else,}\end{cases}$
$\displaystyle(\mathcal{M}_{2,t}f)(x_{1},x_{2})$
$\displaystyle=\begin{cases}0&\text{if }x_{1}\geq 0\text{ and }x_{2}\geq-t,\\\
f(x_{1},x_{2}+t)&\text{else.}\end{cases}$

Then $(\mathcal{M}_{1},\mathcal{M}_{2})$ is a pair of commuting semigroups of
isometries on $\mathcal{H},$ where $\mathcal{M}_{1}=(\mathcal{M}_{1,t})_{t\geq
0}$ and $\mathcal{M}_{2}=(\mathcal{M}_{2,s})_{s\geq 0}.$ Let $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{S}_{j,t}\mkern-1.5mu}\mkern
1.5mu:L^{2}(\mathbb{R}^{2})\to L^{2}(\mathbb{R}^{2})$ for
$j=1,2,t\in\mathbb{R}$ be defined by

$\displaystyle(\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{S}_{1,t}\mkern-1.5mu}\mkern
1.5muf)(x_{1},x_{2})=f(x_{1}-t,x_{2}),\quad(\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{S}_{2,t}\mkern-1.5mu}\mkern
1.5muf)(x_{1},x_{2})=f(x_{1},x_{2}-t).$

for $f\in L^{2}(\mathbb{R}^{2})$ and $(x_{1},x_{2})\in\mathbb{R}^{2}.$ Note
that $({(\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{S}_{1}\mkern-1.5mu}\mkern
1.5mu)}^{*},{(\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{S}_{2}\mkern-1.5mu}\mkern 1.5mu)}^{*})$ on
$L^{2}(\mathbb{R}^{2})$ is the minimal unitary extension of
$(\mathcal{M}_{1},\mathcal{M}_{2}).$ Therefore the dual of
$(\mathcal{M}_{1},\mathcal{M}_{2})$ is $(\mathcal{S}_{1},\mathcal{S}_{2})$ on
$L^{2}(\mathbb{R}_{+}^{2}).$ Hence $(\mathcal{M}_{1},\mathcal{M}_{2})$ is dual
doubly commuting.

A pair $(V_{1},V_{2})$ of commuting semigroups of isometries on $\mathcal{H}$
is called completely nonunitary (c.n.u) if there is no non-zero reducing
subspace $\mathcal{H}_{0}$ of $\mathcal{H}$ for both $V_{1}$ and $V_{2}$ so
that $V_{i}|_{\mathcal{H}_{0}}$ is a unitary semigroup for $i=1,2.$

######  Remark 4.2.

  1. (1)

1.1 for a semigroup $V=(V_{t})_{t\geq 0}$ of isometries, in particular, gives
us the Wold decomposition of $V_{t}$ for each $t>0.$

  2. (2)

Let $(v_{1},v_{2})$ be a pair of commuting isometries. Let $\mathcal{H}_{u}$
be the unitary part in the Wold decomposition of $v=v_{1}v_{2}.$ Then
$\mathcal{H}_{u}$ is reducing for both $v_{1}$ and $v_{2}.$

  3. (3)

Using the above, it is easy to notice that $(V_{1},V_{2})$ is a c.n.u pair of
semigroups of isometries if and only if the product semigroup
$V=V_{1}V_{2}=(V_{1,t}V_{2,t})_{t\geq 0}$ is c.n.u.

This section uses the ideas from [9] extensively.

######  Lemma 4.3.

The dual pair $(\widetilde{V_{1}},\widetilde{V_{2}})$ is always c.n.u.

###### Proof.

We have $\mkern 1.5mu\overline{\mkern-1.5muV_{i,t}\mkern-1.5mu}\mkern
1.5mu=\begin{pmatrix}V_{i,t}&\star\\\
0&(\widetilde{V_{i,t}})^{*}\end{pmatrix}$ on $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu=\mathcal{H}\oplus\widetilde{\mathcal{H}}$ for $t\geq 0,i=1,2.$ By
minimality, $\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu=\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu(\mathcal{H}):t,s\in\mathbb{R}\\}.$ Let
$\widetilde{V}=\widetilde{V_{1}}\widetilde{V_{2}},$ that is,
$\widetilde{V_{t}}=\widetilde{V_{1,t}}\widetilde{V_{2,t}}$ for all $t\geq 0.$
Let $\mathcal{H}_{0}$ be the unitary part in the Cooper’s decomposition (1.1)
of $\widetilde{V}.$ Suppose $\widetilde{V}$ is not c.n.u. Then
$\mathcal{H}_{0}$ is a non-zero subspace of $\widetilde{\mathcal{H}}$ and
$\mathcal{H}_{0}$ is reducing for $\widetilde{V_{i,t}}$ for all $t\geq
0,i=1,2$ (from Remark 4.2). Since $(\mkern
1.5mu\overline{\mkern-1.5muV_{i,t}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\mathcal{H}_{0}}=\widetilde{V_{i,t}}|_{\mathcal{H}_{0}},$ note
that $\mathcal{H}_{0}$ is a reducing subspace for $\mkern
1.5mu\overline{\mkern-1.5muV_{i,t}\mkern-1.5mu}\mkern 1.5mu$ for all $t\geq 0$
and $i=1,2.$ Therefore,

$\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu(\mathcal{H}):t,s\in\mathbb{R}\\}\perp\mathcal{H}_{0}.$

This is a contradiction to the minimality. ∎

######  Theorem 4.4.

Let $(V_{1},V_{2})$ be a c.n.u pair of commuting semigroups of isometries on
$\mathcal{H}.$ If the pair $(\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu,\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern 1.5mu)$ on $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern 1.5mu$ is the
minimal unitary extension of $(V_{1},V_{2}),$ then the pair $((\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu)^{*},(\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern 1.5mu)^{*})$ is the
minimal unitary extension of the dual $(\widetilde{V_{1}},\widetilde{V_{2}}).$
In particular,
$({\overset{\approx}{V_{1}}},{\overset{\approx}{V_{2}}})=(V_{1},V_{2}),$ where
$({\overset{\approx}{V_{1}}},{\overset{\approx}{V_{2}}})$ is the dual of
$(\widetilde{V_{1}},\widetilde{V_{2}}).$

###### Proof.

Let $\widetilde{\mathcal{H}}=\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu\ominus\mathcal{H}.$ Only minimality of the extension needs to be proved,
i.e.,

$\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu(\widetilde{\mathcal{H}}):t,s\in\mathbb{R}\\}=\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern 1.5mu.$

To that end, let $x\in\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern 1.5mu$ and
$x\perp\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu(\widetilde{\mathcal{H}}):t,s\in\mathbb{R}\\}.$ Then $\langle\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu\tilde{h},x\rangle=0$ for all $\tilde{h}\in\widetilde{\mathcal{H}}$ and
$t,s\in\mathbb{R}.$ This implies $\langle\tilde{h},\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern 1.5mux\rangle=0$ for all
$\tilde{h}\in\widetilde{\mathcal{H}}$ and $t,s\in\mathbb{R}.$ Hence $\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern 1.5mux\in\mathcal{H}$
for all $t,s\in\mathbb{R}.$ So, $X=\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mux:t,s\in\mathbb{R}\\}\subseteq\mathcal{H}.$ Clearly $X$ reduces both
$\mkern 1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu$ and $\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern 1.5mu$ to unitary
semigroups. Since $X\subseteq\mathcal{H}$ we have $\mkern
1.5mu\overline{\mkern-1.5muV_{i,t}\mkern-1.5mu}\mkern 1.5mu|_{X}=V_{i,t}|_{X}$
for $i=1,2$ and for all $t\geq 0.$ This shows that $X$ reduces both $V_{1}$
and $V_{2}$ to unitary semigroups. Now since $(V_{1},V_{2})$ is c.n.u, $X$ has
to be the zero space. In particular, $x=0.$ This shows the minimality. ∎

######  Definition 4.5.

A c.n.u pair $(V_{1},V_{2})$ of commuting semigroups of isometries is called a
modified-bishift-semigroup if the dual $(\widetilde{V_{1}},\widetilde{V_{2}})$
of $(V_{1},V_{2})$ is a bishift-semigroup.

The following is the main result of this section. It gives a Cooper type
decomposition for the pairs of dual doubly commuting semigroups of isometries.

######  Theorem 4.6.

Let $(V_{1},V_{2})$ be a pair of commuting semigroups of isometries on
$\mathcal{H}$. Suppose $(V_{1},V_{2})$ is dual doubly commuting. Then the
space $\mathcal{H}$ decomposes as

$\mathcal{H}=\mathcal{H}_{m}\oplus\mathcal{H}_{p,u}\oplus\mathcal{H}_{u,p}\oplus\mathcal{H}_{u,u},$

so that $\mathcal{H}_{m},\mathcal{H}_{p,u},\mathcal{H}_{u,p}$ and
$\mathcal{H}_{u,u}$ reduce both $V_{1}$ and $V_{2},$
(1) $(V_{1},V_{2})$ is a modified-bishift-semigroup on $\mathcal{H}_{m},$
(2) $V_{1}$ is a c.n.u semigroup, $V_{2}$ is a unitary semigroup on
$\mathcal{H}_{p,u},$
(3) $V_{1}$ is a unitary semigroup, $V_{2}$ is a c.n.u semigroup on
$\mathcal{H}_{u,p},$ and
(4) both $V_{1}$ and $V_{2}$ are unitary semigroups on $\mathcal{H}_{u,u}.$

###### Proof.

First assume that $(V_{1},V_{2})$ is a c.n.u pair of dual doubly commuting
semigroups of isometries. Let $(\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu,\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern 1.5mu)$ on $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern 1.5mu$ be the
minimal unitary extension of $(V_{1},V_{2})$ and let
$(\widetilde{V_{1}},\widetilde{V_{2}})$ be the dual of $(V_{1},V_{2}).$

Since $(V_{1},V_{2})$ is dual doubly commuting,
$(\widetilde{V_{1}},\widetilde{V_{2}})$ is doubly commuting. Hence by 3.3 and
3.5 we have
$\widetilde{\mathcal{H}}=\widetilde{\mathcal{H}}_{p,p}\oplus\widetilde{\mathcal{H}}_{p,u}\oplus\widetilde{\mathcal{H}}_{u,p}\oplus\widetilde{\mathcal{H}}_{u,u},$
where $(\widetilde{V_{1}},\widetilde{V_{2}})$ is a bishift-semigroup on
$\widetilde{\mathcal{H}}_{p,p},$ $\widetilde{V_{1}}$ is a c.n.u semigroup and
$\widetilde{V_{2}}$ is a unitary semigroup on $\widetilde{\mathcal{H}}_{p,u},$
and $\widetilde{V_{1}}$ is a unitary semigroup and $\widetilde{V_{2}}$ is a
c.n.u semigroup on $\widetilde{\mathcal{H}}_{u,p}.$ By Lemma 4.3 we have
$\widetilde{\mathcal{H}}_{u,u}=\\{0\\}.$

For $i,j\in\\{p,u\\}$ and $(i,j)\neq(u,u),$ let

$\hat{\mathcal{H}}_{i,j}=\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muV_{2,s}\mkern-1.5mu}\mkern
1.5mu(\widetilde{\mathcal{H}}_{i,j}):t,s\in\mathbb{R}\\}.$

Then, it is not difficult to show that
$\hat{\mathcal{H}}_{p,p},\hat{\mathcal{H}}_{p,u}$ and
$\hat{\mathcal{H}}_{u,p}$ are pairwise orthogonal and $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{H}\mkern-1.5mu}\mkern
1.5mu=\hat{\mathcal{H}}_{p,p}\oplus\hat{\mathcal{H}}_{p,u}\oplus\hat{\mathcal{H}}_{u,p}.$
Also clearly $\hat{\mathcal{H}}_{i,j}$ is a reducing subspace for $\mkern
1.5mu\overline{\mkern-1.5muV_{k,t}\mkern-1.5mu}\mkern 1.5mu$ for $k=1,2,t\geq
0$ and $i,j\in\\{p,u\\},(i,j)\neq(u,u).$

Let
$\mathcal{H}_{m}=\hat{\mathcal{H}}_{p,p}\ominus\widetilde{\mathcal{H}}_{p,p}$
and
$\mathcal{H}_{i,j}=\hat{\mathcal{H}}_{i,j}\ominus\widetilde{\mathcal{H}}_{i,j}$
for $i,j\in\\{p,u\\}$ with $i\neq j.$ Then

$\mathcal{H}=\mathcal{H}_{m}\oplus\mathcal{H}_{p,u}\oplus\mathcal{H}_{u,p}.$
(4.1)

Note that $\mathcal{H}_{m},\mathcal{H}_{p,u}$ and $\mathcal{H}_{u,p}$ are
invariant subspaces for $\mkern
1.5mu\overline{\mkern-1.5muV_{k,t}\mkern-1.5mu}\mkern 1.5mu$ for all $t\geq
0,k=1,2.$ Hence they are invariant subspaces for $V_{k,t}.$ Thus by Eq. 4.1,
they are reducing subspaces for $V_{k,t}.$

Note that the pair $(\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern
1.5mu|_{\hat{\mathcal{H}}_{p,p}},\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu|_{\hat{\mathcal{H}}_{p,p}})$ on $\hat{\mathcal{H}}_{p,p}$ is the minimal
unitary extension for $(V_{1}|_{\mathcal{H}_{m}},V_{2}|_{\mathcal{H}_{m}})$ on
$\mathcal{H}_{m}$ and the pair $(\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern
1.5mu|_{\hat{\mathcal{H}}_{i,j}},\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu|_{\hat{\mathcal{H}}_{i,j}})$ on $\hat{\mathcal{H}}_{i,j}$ is the minimal
unitary extension for
$(V_{1}|_{\mathcal{H}_{i,j}},V_{2}|_{\mathcal{H}_{i,j}})$ on
$\mathcal{H}_{i,j}$ for $i,j\in\\{p,u\\}$ and $i\neq j.$

Now we shall show that $(V_{1}|_{\mathcal{H}_{m}},V_{2}|_{\mathcal{H}_{m}})$
is a modified-bishift-semigroup on $\mathcal{H}_{m}.$ To show that, we must
show that $((\mkern 1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\widetilde{\mathcal{H}}_{p,p}},(\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\widetilde{\mathcal{H}}_{p,p}})$ is a bishift-semigroup on
$\widetilde{\mathcal{H}}_{p,p}.$ For this, note that $((\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\widetilde{\mathcal{H}}_{p,p}},(\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\widetilde{\mathcal{H}}_{p,p}})=(\widetilde{V_{1}}|_{\widetilde{\mathcal{H}}_{p,p}},\widetilde{V_{2}}|_{\widetilde{\mathcal{H}}_{p,p}})$
and it is a bishift-semigroup.

Next we shall show that $V_{1}|_{\mathcal{H}_{p,u}}$ is a c.n.u semigroup and
$V_{2}|_{\mathcal{H}_{p,u}}$ is a unitary semigroup on $\mathcal{H}_{p,u}.$
Note that $(\mkern 1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\hat{\mathcal{H}}_{p,u}}$ is a unitary semigroup on
$\hat{\mathcal{H}}_{p,u}$ and $(\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu)^{*}|_{\widetilde{\mathcal{H}}_{p,u}}=\widetilde{V_{2}}|_{\widetilde{\mathcal{H}}_{p,u}}$
is a unitary semigroup on $\widetilde{\mathcal{H}}_{p,u}.$ Hence $\mkern
1.5mu\overline{\mkern-1.5muV_{2}\mkern-1.5mu}\mkern
1.5mu|_{\mathcal{H}_{p,u}}=V_{2}|_{\mathcal{H}_{p,u}}$ is a unitary semigroup
on $\mathcal{H}_{p,u}.$ Note that

$\hat{\mathcal{H}}_{p,u}=\overline{\operatorname{span}}\\{\mkern
1.5mu\overline{\mkern-1.5muV_{1,t}\mkern-1.5mu}\mkern
1.5mu(\widetilde{\mathcal{H}}_{p,u}):t\in\mathbb{R}\\}.$ (4.2)

Since $\mkern 1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern 1.5mu^{*}$
is a unitary extension for the c.n.u semigroup
$\widetilde{V_{1}}|_{\widetilde{\mathcal{H}}_{p,u}}$ on
$\widetilde{\mathcal{H}}_{p,u},$ by Eq. 4.2 one can see that
$\hat{\mathcal{H}}_{p,u}$ is the minimal space for this extension. Therefore,
since
$\mathcal{H}_{p,u}=\hat{\mathcal{H}}_{p,u}\ominus\widetilde{\mathcal{H}}_{p,u},$
we see (using the same techniques as in Lemma 4.3) that $\mkern
1.5mu\overline{\mkern-1.5muV_{1}\mkern-1.5mu}\mkern
1.5mu|_{\mathcal{H}_{p,u}}=V_{1}|_{\mathcal{H}_{p,u}}$ is also a c.n.u
semigroup on $\mathcal{H}_{p,u}.$

Similarly we can show that $V_{1}|_{\mathcal{H}_{u,p}}$ is a unitary
semigroup, and $V_{2}|_{\mathcal{H}_{u,p}}$ is a c.n.u semigroup on
$\mathcal{H}_{u,p}.$ This proves the theorem for a c.n.u pair $(V_{1},V_{2}).$

Now let $(V_{1},V_{2})$ be any pair (not necessarily c.n.u) of dual doubly
commuting semigroups of isometries. Let $\mathcal{H}_{u,u}$ be the unitary
part in 1.1 for $V_{1}V_{2}.$ Then $\mathcal{H}_{u,u}$ reduces both $V_{1}$
and $V_{2},$ hence $V_{1}|_{\mathcal{H}_{u,u}}$ and
$V_{2}|_{\mathcal{H}_{u,u}}$ are unitary semigroups.

Let $\mathcal{H}_{s}=\mathcal{H}\ominus\mathcal{H}_{u,u}.$ Since the dual of
$(V_{1},V_{2})$ is same as the dual of
$(V_{1}|_{\mathcal{H}_{s}},V_{2}|_{\mathcal{H}_{s}}),$
$(V_{1}|_{\mathcal{H}_{s}},V_{2}|_{\mathcal{H}_{s}})$ is a c.n.u pair of dual
doubly commuting semigroups of isometries. Hence by applying the above proof
to the c.n.u pair $(V_{1}|_{\mathcal{H}_{s}},V_{2}|_{\mathcal{H}_{s}}),$ we
get the proof of the theorem. ∎

The converse of this theorem also holds and it is easy to see. Note that the
dual of a commuting pair of a c.n.u semigroup and a unitary semigroup is again
a commuting pair of a c.n.u semigroup and a unitary semigroup from Corollary
3.7.

The next result shows that $(\mathcal{M}_{1},\mathcal{M}_{2})$ of Example 4.1
is a model for the pairs of dual doubly commuting semigroups of isometries.

######  Theorem 4.7.

Let $(V_{1},V_{2})$ be a modified-bishift-semigroup on $\mathcal{H}.$ Then
$(V_{1},V_{2})$ is jointly unitarily equivalent to $(\mathcal{M}_{1}\otimes
I_{\mathcal{F}},\mathcal{M}_{2}\otimes I_{\mathcal{F}})$ for some Hilbert
space $\mathcal{F}.$

###### Proof.

Since the dual $(\widetilde{V_{1}},\widetilde{V_{2}})$ of $(V_{1},V_{2})$ is a
bishift-semigroup, $(\widetilde{V_{1}},\widetilde{V_{2}})$ is jointly
unitarily equivalent to $(\mathcal{S}_{1}\otimes
I_{\mathcal{F}},\mathcal{S}_{2}\otimes I_{\mathcal{F}})$ for some Hilbert
space $\mathcal{F}.$ Note that the dual of $(\mathcal{S}_{1}\otimes
I_{\mathcal{F}},\mathcal{S}_{2}\otimes I_{\mathcal{F}})$ is
$(\mathcal{M}_{1}\otimes I_{\mathcal{F}},\mathcal{M}_{2}\otimes
I_{\mathcal{F}}),$ hence by 4.4, $(V_{1},V_{2})$ is jointly unitarily
equivalent to $(\mathcal{M}_{1}\otimes I_{\mathcal{F}},\mathcal{M}_{2}\otimes
I_{\mathcal{F}}).$ ∎

The forward part of the following corollary follows from 4.6, 4.7 and the fact
that $(\mathcal{M}_{1},\mathcal{M}_{2})$ is not doubly commuting. The converse
part follows from [7] and 4.6.

######  Corollary 4.8.

Let $(V_{1},V_{2})$ be a pair of commuting semigroups of isometries on
$\mathcal{H}.$ Then $(V_{1},V_{2})$ is simultaneously doubly commuting and
dual doubly commuting if and only if $\mathcal{H}$ decomposes as the direct
sum

$\mathcal{H}=\mathcal{H}_{p,u}\oplus\mathcal{H}_{u,p}\oplus\mathcal{H}_{u,u}$

of reducing subspaces $\mathcal{H}_{p,u},\mathcal{H}_{u,p}$ and
$\mathcal{H}_{u,u}$ for both $V_{1}$ and $V_{2},$ such that
(1) $V_{1}$ is a c.n.u semigroup and $V_{2}$ is a unitary semigroup on
$\mathcal{H}_{p,u},$
(2) $V_{1}$ is a unitary semigroup and $V_{2}$ is a c.n.u semigroup on
$\mathcal{H}_{u,p},$ and
(3) both $V_{1}$ and $V_{2}$ are unitary semigroups on $\mathcal{H}_{u,u}.$

Acknowledgements The authors thank the referee for a careful reading and
useful suggestions. Research is funded by the J C Bose fellowship
JCB/2021/000041, the D S Kothari postdoctoral fellowship MA/20-21/0047 and the
DST FIST program - 2021 [TPN - 700661].

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A fine-grained parallelization of the immersed boundary method

Andrew Kassen1,
Varun Shankar2 and
Aaron L. Fogelson1,3

1Department of Mathematics, University of Utah, Salt Lake City, UT, USA

2School of Computing, University of Utah, Salt Lake City, UT, USA

3Department of Biomedical Engineering, University of Utah, Salt Lake City, UT, USA

Aaron L. Fogelson, 155 S 1400 E, JWB 233, Salt Lake City, UT 84112, USA

We present new algorithms for the parallelization of Eulerian-Lagrangian interaction
operations in the immersed boundary method. Our algorithms rely on two well-studied
parallel primitives: key-value sort and segmented reduce. The use of these parallel
primitives allows us to implement our algorithms on both graphics processing units (GPUs)
and on other shared memory architectures. We present strong and weak scaling tests on
problems involving scattered points and elastic structures. Our tests show that our
algorithms exhibit near-ideal scaling on both multicore CPUs and GPUs.

§ INTRODUCTION

Many problems in biophysics involve the interaction of an incompressible fluid and an
immersed elastic interface. The solution to these problems can be approximated using the
immersed boundary (IB) method, which was developed in [Peskin, 1972] to simulate
blood flow through the heart. The relative ease by which the IB method can be
incorporated into a Navier-Stokes solver has led to its popularity in myriad applications
(see, e.g. [Mittal and Iaccarino, 2005, Griffith and Patankar, 2020] and references therein). The IB method
couples the equations governing fluid velocity and pressure to those governing the
interface movement and elastic forces via operations we will refer to as
interpolation and spreading. Fluid velocities are interpolated to points on
the interface, and forces on the interface are spread to the fluid. Fluid properties are
discretized on a fixed Eulerian grid. The interface is represented by a set of mobile
Lagrangian points. This presents a problem for effective parallelization of interpolation
and spreading, as these operators must be reconstructed at each timestep to account for
the motion of the Lagrangian points relative to the Eulerian grid. The Eulerian and
Lagrangian points, on the other hand, can be treated as fixed in their respective
coordinate spaces for several timesteps, if not for the entirety of a simulation. We may
therefore treat solving the fluid equations and computing elastic forces as the result of
a black-box computation and focus primarily on the interpolation and spreading

We are interested in the case of neutrally buoyant, elastic immersed structures that move
at the local fluid velocity. In particular, we aim to simulate whole blood, which is
composed of red blood cells (RBCs) and platelets immersed in blood plasma, within a
vessel lined by endothelial cells. The cells are elastic, and as they deform with the
flow of the enveloping fluid, they impose a force on the fluid. Approximately 40% of
the volume in healthy human blood is occupied by RBCs, and domains containing just a few
cells may require tens or hundreds of thousands of points to discretize these cells for
use within the IB method.

To take advantage of modern computing architectures, with ever-increasing numbers of
processors, it is necessary to develop parallel algorithms for the IB method.
[McQueen and Peskin, 1997] present a domain decomposition scheme to parallelize the
interpolation and spreading operations on the Cray C-90 computer with shared memory and a
modest number of vector processors. Their results illustrate the need for a fast
interpolation and spreading: even parallelized, they spend roughly half of the wall clock
time spreading and interpolating. [Fai et al., 2013] adapted this domain decomposition
scheme for use on a general purpose graphical processing unit (GPGPU; GPU for short).
[Patel, 2012] parallelized spreading on a GPU by processing one Lagrangian point at
a time. Because neither of these approaches can concurrently process arbitrary Lagrangian
points, it is easy to find cases for which they perform poorly. An alternative is to
distribute work among several devices, each with its own memory. This idea underpins the
popular IBAMR library (see [Griffith et al., 2010, Griffith et al., 2007, Griffith, 2009, Griffith, 2011, Griffith and Luo, 2017]), which also adaptively refines the mesh around the
immersed structure. With proper load-balancing, this allows a serial algorithm to process
a smaller portion of work. A cluster of multicore devices, however, will not be used
effectively without a shared-memory parallel algorithm. The cuIBM ([Layton et al., 2011])
and PetIBM ([Mesnard and Barba, 2017, Chuang et al., 2018]) libraries implement an adaptive IB
method for prescribed motion on single- and multi-GPU architectures, respectively. The
authors demonstrate their method on a few two-dimensional test problems. Their
implementation explicitly constructs the spreading and interpolation operators, which are
sparse. The interpolation matrix has approximately the same number of nonzero
entries per row, can be constructed in parallel in compressed sparse row (CSR) format,
and its multiplication against a column vector parallelizes easily as a series of sparse
dot products. The spreading matrix, with approximately the same number of nonzero
entries per column, can be constructed in parallel in compressed sparse column (CSC)
format, for which parallel multiplication against a column vector typically requires
conversion to a different format. Even after conversion to CSR, the spreading matrix does
not have approximately the same number of entries per row so does not always yield
effective parallelization of matrix-vector products.

GPUs have restrictions on their parallelization. They use single instruction,
multiple data (SIMD) parallelism, in which each computational unit, or thread, executes
the same instruction on its own data. Concurrency on the GPU is typically limited by the
amount of shared memory, which is shared among a group of threads, and register memory,
which is accessible only to a single thread. The alternative is to use global memory,
which is slow in general, but faster when accesses are sequential (or “coalesced”).
These restrictions on the GPU imply that an effective algorithm for the GPU translates
well to other shared-memory architectures, such such as multicore CPUs, which support
parallelism, SIMD instructions, advanced instruction pipelining, and out-of-order
execution. We therefore develop parallel spreading and interpolation algorithms
applicable to both GPUs and multicore CPUs. The success of our new algorithms relies in
dividing these operations into trivially parallelizable tasks and parallel primitives.

The remainder of this paper is organized into four sections. Section <ref> gives
an overview of the IB method, and describes the role of the interpolation and spread
operations. In Section <ref>, we discuss the parallelization of
interpolation and introduce a new parallelization for the spreading operation. In
Section <ref>, we demonstrate that the concurrency of these algorithms scales
with the number of IB points, independently of the Eulerian grid. We confirm the
convergence of the IB method with these new algorithms. We illustrate the suitability of
our algorithms to both GPUs and multicore CPU architectures with both weak and strong
scaling tests. Finally, we summarize our findings in Section <ref>.

§ REVIEW OF THE IMMERSED BOUNDARY METHOD

Consider a $d$-dimensional ($d=2$ or 3) rectangular domain $\domain$, which is filled
with a viscous incompressible fluid with constant viscosity $\mu$ and density $\rho$, and
contains an immersed elastic structure, $\interface$. The structure is impermeable to the
fluid and moves at the local fluid velocity, is deformed by this motion, and imparts a
force to the fluid. Otherwise, the interface is treated as part of the fluid.

The fluid velocity, $\u = \u(\x,\,t)$, and pressure, $p = p(\x,\,t)$, are governed by the
incompressible Navier-Stokes equations for a Newtonian fluid,

\begin{gather}
\label{eq:ins-evol}
\rho(\u_t + \u\cdot\grad\u) = \mu\Delta\u - \grad p + \f, \\
\label{eq:ins-incomp}
\div\u = 0,
\end{gather}

where $\f$ is the elastic force density. This is a set of $d+1$ equations in $d+1$
unknowns: the $d$ components of $\u$, and $p$. The equations are written relative to the
Eulerian frame, so that the coordinates $$ are independent variables. Throughout, we
write Eulerian quantities in the lowercase Latin alphabet.

Let $=(, t)$ represent a parametrization of the Cartesian coordinates of the
immersed interface with material coordinates $$ at time $t$. Let $Ł[]$ be the
energy density functional for the elastic interface material. The elastic force density
is computed by evaluating% the Fréchet derivative of $\L$,
\begin{equation}
\label{eq:forces}
\F = -\delta \L[\X],
\end{equation}
where $δ$ represents the first variation. Uppercase Latin letters represent
Lagrangian quantities and are functions of $$ and $t$.

The interface moves at the local fluid velocity, and force balance on the interface
between the interface and fluid dictates that the interface force on the fluid is equal
to the elastic force. Analytically, the fluid-interface interactions can be written
\begin{gather}
\label{eq:interpolation}
\U(\params,\,t) = \int_\domain \Dirac(\x-\X(\params,\,t)) \u(\x,\,t)\d\x,\ \text{and} \\
\label{eq:spreading}
\f(\x,\,t) = \int_\interface \Dirac(\x-\X(\params,\,t))\F(\params,\,t)\d\params,
\end{gather}
where $(, t)$ represents the derivative of $(, t)$ with respect to
$t$, and $(-(, t))$ is the Dirac $$-function centered at
$(, t)$. Equation~\eqref{eq:interpolation} is called \term{interpolation}, and
the result of the right-hand side is the fluid velocity at $(, t)$; namely,
$(, t) = (̆(, t), t)$. Equation~\eqref{eq:spreading} is called
\term{spreading}. {\reva$\F(\params,\,t)$ and $\f(\x,\,t)$ may not have the same units, but
the force $\F(\params,\,t)\d\params$ over $\d\params$ is ``spread'' to the force
$\f(\x,\,t)\d\x$ over $\d\x$.}

The fluid equations~\eqref{eq:ins-evol}--\eqref{eq:ins-incomp} are spatially discretized
on a regular background grid of spacing $h$ so that $$ is divided into square or
cubic cells of side length $h$. Because of the checkerboard instability (see, e.g.,~%
[Wesseling, 2001]) in solving the Navier-Stokes equations on collocated regular
grids, {\revab we use a MAC grid ([Harlow and Welch, 1965]) to stagger the components of Eulerian
vector quantities. For example, if $\vec{v}$ is an Eulerian vector-valued function,
$\e$ is a canonical basis vector, and $\x$ is a cell center, $\e\cdot\vec{v}$ is
discretized at $\x - h\e/2$. These discretization points comprise a new, regular
grid with spacing $h$. For the component under consideration, $\domain_h$ denotes the set
of discretization points with cardinality $n_\omega$.} The momentum equation~%
\eqref{eq:ins-evol} is discretized in time using a {\reva first-order method based on
the} implicit-explicit Runge-Kutta method described in~[Peskin, 2002], and the
incompressibility condition~\eqref{eq:ins-incomp} is satisfied using PmIII of~%
[Brown et~al., 2001].

The Lagrangian force density~\eqref{eq:forces} is evaluated at a set of points, usually
{\revb with \emph{fixed} material coordinates}, referred to as IB points. The notation
$_j=(_j, t)$ refers to an individual IB point on $$ in Cartesian
coordinates. The typical heuristic for distributing the points $_j$ on a connected
interface places neighboring IB points at most $h$ apart from one another, and often at
most $h/2$ apart. We therefore denote the set of IB points by $_h$ and the
number of IB points by $n_γ$. From these points, we construct a smooth approximation
to the interface using radial basis functions (RBFs), as in~[Shankar et~al., 2015]. This
allows us to calculate geometric properties using well-known formulas, and evaluate the
forces~\eqref{eq:forces} in analytic form, similar to that of~[Maxian et~al., 2018].

The singular integrals~\eqref{eq:interpolation} and~\eqref{eq:spreading} do not lend
themselves easily to evaluation. In particular, it is unlikely that IB points and grid
points will coincide. For a regular grid with spacing $h$, we replace the Dirac
$$-function with a regularized kernel, $_h$, which is a product of
one-dimensional kernels, $h^-1(h^-1x)$.~[Griffith and Patankar, 2020] gives several
options for $$. {\revb The methods described in this paper do not depend on a particular
$\kernel$, but we restrict ourselves to the cosine kernel from~[Peskin, 2002].
The cosine kernel is simple to implement, but does not satisfy all of the moment
conditions for a 4-point kernel, though it approximates a kernel with the appropriate
discrete properties.}

Let $_h$ be one of the Eulerian grids for vector-valued quantities. {\revb Discretizing
the component of equations~\eqref{eq:interpolation} and~\eqref{eq:spreading} on
$\domain_h$ and $\interface_h$, respectively, corresponding to canonical basis vector
$\e$ yields
\begin{align}
\label{eq:disc-interp}%
\e\cdot\U^n_j &= \sum_{i=1}^{n_\omega} \delta_h(\x_i-\X^n_j)\e\cdot\u^n_i h^d \quad \text{and} \\
\label{eq:disc-spread}%
\e\cdot\f^n_i &= \sum_{j=1}^{n_\gamma} \delta_h(\x_i-\X^n_j)\e\cdot\F^n_j \d\theta_j,
\end{align}
where $\u^n_i$ and $\f^n_i$ are discrete approximations to $\u$ and $\f$}
at $\x_i\in\domain_h$ and time $t=t_n$, and {\revb$\U^n_{\smash j}$} and $\F^n_{\smash j}$
are their Lagrangian counterparts at $\X_{\smash j}$, respectively. The terms $h^d$ and
$\d\theta_{\smash j}$ are integration weights analogous to $\d\x$ and $\d\params$. For
a topologically spherical interface, we compute weights on the unit sphere using RBFs,
using a simplified version of the method described in~[Fuselier et~al., 2013], and use
the Jacobian of the mapping between the sphere and interface to obtain
$\d\theta_{\smash j}$. Both~\eqref{eq:disc-interp} and~\eqref{eq:disc-spread} look like a
matrix-vector multiplication, so we define the \emph{spreading matrix}
$\spread=(\delta_h(\x_i-\X^n_{\smash j}))$ with row $i$ and column $j$. We call its
transpose, $\interp$, the \emph{interpolation matrix}. Collecting values
{\reva $\e\cdot\u^n_ih^d$ at each grid point as $\vec{v}^n$ and
$\e\cdot\F^n_{\smash j}\d\theta_{\smash j}$ at each IB point as $\vec{G}^n$}, we
rewrite equations~\eqref{eq:disc-interp} and~\eqref{eq:disc-spread} in matrix form as
\begin{align}
\label{eq:matrix-interp}%
\U^n &= \interp{\vec{v}}^n \quad\text{and} \\
\label{eq:matrix-spread}%
\f^n &= \spread{\vec{G}}^n,
\end{align}}

Collectively, the equations~\eqref{eq:ins-evol}--\eqref{eq:spreading} constitute the
IB method, and while different implementations {\reva of the IB method use different
spatial and temporal discretizations, surface representations, elastic models, or choice
of $\delta_h$}, a single step proceeds with timestep $k$ roughly as follows:
\begin{enumerate}[label=(\texttt{\alph*}), itemsep=0ex]
\item interpolate $\u^n$ to $\X^n$ to get {\revb $\U^\ast$},
\item predict Lagrangian positions $\X^\ast = \X^n + k{\revb \U^\ast}$,
\item compute Lagrangian forces $\F^\ast$ using positions $\X^\ast$,
\item spread {\reva$\F^\ast$} from $\X^\ast$ to get $\f^\ast$,
\item solve for updated Eulerian velocities $\u^{n+1}$,
\item interpolate $\u^{n+1}$ to $\X^n$ to get {\revb$\U^{n+1}$}, and
\item update $\X^{n+1} = \X^n + k {\revb\U^{n+1}}$.
\end{enumerate}
We group these steps into 3 categories: the purely Eulerian (\texttt{e}); the purely
Lagrangian (\texttt{b}), (\texttt{c}), and (\texttt{g}); and the Eulerian-Lagrangian
coupling (\texttt{a}), (\texttt{d}), and (\texttt{f}). We discuss the first two
categories in a forthcoming paper. The remainder of this paper discusses the
implementation and parallelization of the third.

% vim: cc=90 tw=89

\section{Parallelization strategies for interpolation and spreading} \label{sec:parallel}

Here, we describe a method for evaluating the discrete counterparts to
\begin{align}
\label{eq:scalar-interp}
E(\X) &= \int_\domain \Dirac_h(\x-\X)e(\x) \d\omega \quad \text{and}\\
\label{eq:scalar-spread}
\ell(\x) &= \int_\interface \Dirac_h(\x-\X)L(\X) \d\gamma
\end{align}
where scalar-valued Eulerian function $e:\domain\to\mathbb{R}$ is interpolated to
Lagrangian point $\X$ and Lagrangian function $L:\interface\to\mathbb{R}$ is spread to
Eulerian point $\x$. For vector-valued functions, such as the Eulerian fluid velocity
$\u$ and Lagrangian force density $\F$, the algorithm can be applied to each component
individually. For a staggered grid, this is necessary, as the grid for each component has
a different set of grid points, and may have a different number of grid points. In the
following, we discuss parallelizing equation \eqref{eq:scalar-interp} and introduce a
novel parallelization scheme for evaluating equation \eqref{eq:scalar-spread}. We begin
by defining some notation that will be used throughout the description of these

\input{ib-discretization}
\input{spread-serial}
\input{interp-parallel}
\input{spread-parallel}

% vim: cc=90 tw=89

\section{Numerical results} \label{sec:results}

Here we describe two types of test: unstructured IB points, in which points are placed
randomly in the domain and generate a force independently from the other IB points, and
structured IB points, in which the points comprise an elastic structure and forces are
generated based on the configuration of the points as a whole. For these tests, we use
a $16\um\times16\um\times16\um$ triply periodic domain with an initially shear-like flow,
$\u=(0,\,0,\,\shear(y-8\um))$, with shear rate $\shear$. Tests use a shear rate of
$1000\si{\per\second}$ unless otherwise noted. This flow has a sharp transition at the
periodic boundary $y=0\um$, so a background force is added to maintain this transition
and so that the initial flow is also the steady flow in the absence of other forces.

Serial and multicore CPU tests were performed on a single node with 48 Intel{\reg}
Xeon{\reg} CPU E5-2697 v2 2.70\si{\giga\hertz} processors and 256 GB of RAM running
CentOS Linux release 7.7.1908 (x86\_64). Parallel CPU implementations use Intel's OpenMP
library, \texttt{libiomp5}. GPU tests used the same node with an NVIDIA{\reg} Tesla{\reg}
K80 ($2\times$GK210 GPU with 13 823.5\si{\mega\hertz} multiprocessors and 12 GB of global
memory each). Only one of the GK210 GPUs was used. The CPU code was written in C++20 and
the GPU code was written in C++/CUDA and used version 9.2 CUDA libraries ([Kassen, 2021]).
Both the CPU and GPU code were compiled using \texttt{clang} version 7.0.1. All tests are
performed in double precision. We begin with tests using unstructured IB points, for
which both of these architectures were used.

\subsection{Unstructured IB points}\label{sec:unst}

Consider a set of $n$ IB points randomly placed in the domain described above. The IB
points are imagined to be tethered to their initial positions. The fluid solver is not
invoked for these tests. Instead, at each timestep, we interpolate the fluid velocity to
each of the IB points and predict updated positions for the IB points. Using these
predicted positions, we compute a Hookean force for each IB point with spring constant
$0.01\si{dyn\per\centi\meter}$. We spread these forces from the predicted positions to
the fluid grid, but do not use them to update the fluid velocity. This ensures that the
points do not settle into a steady position so the spreading and interpolation operations
receive new data each timestep. Finally, we interpolate the velocity to the positions of
the IB points at the beginning of the timestep again and update the position of the IB
points. While the interpolated velocities are the same as those computed at the beginning
of the timestep, this is done by analogy to the fluid solver, which interpolates fluid
velocities twice and spreads forces once per timestep.

We use this test to compare the performance of the parallel algorithms to their serial
counterparts and, for the spreading variants, to each other.

\input{grid-dependence}
\input{serial-comparison}

\subsection{Elastic objects}

We are motivated by the desire to simulate the motion of cells immersed in a fluid. Cells
are not randomly generated points, but cohesive structures, kept together by elastic
forces and the near-constant volume enclosed by their membranes. In this section, we
replace the randomly-placed IB points with points sampled on the surface of either a
sphere or an RBC. We track $n_d$ data sites per object, and interpolate fluid velocities
only to these points. We construct an RBF interpolant based on the positions of the data
sites and evaluate forces at $n_s$ sample sites, chosen so that neighboring sample sites
are initially within approximately $0.5h$ of each other. We spread forces from the sample
sites. It is generally the case that $n_s > n_d$, so that we interpolate to fewer points
than we spread from. In the parlance of Section \ref{sec:parallel}, $n_\gamma=n_d$ in the
context of interpolation, and $n_\gamma=n_s$ in the context of spreading. {\revb This violates
the transpose property of Equations~\eqref{eq:matrix-interp}--\eqref{eq:matrix-spread}.
Neither $\spread$ nor $\interp$ are constructed explicitly, so the algorithms presented
above work without modification. However, the $\interface_h$ argument will differ between
the spreading and interpolation algorithms.} Data and sample sites are generated using
the method described by~[Shankar et~al., 2018]. In this case, we invoke the fluid solver,
so that as the object deforms, the force it imparts on the fluid will affect the fluid
velocity. The sphere and RBC are elastic, obeying the Skalak constitutive law
([Skalak et~al., 1973]) with shear modulus $2.5\times10^{-3}\dynpercm$ and bulk modulus
$2.5\times10^{-1}\dynpercm$. The RBC has the reference configuration given in~%
[Omori et~al., 2012]:
\begin{equation}
\begin{aligned}
x(\theta,\,\varphi) &= R_0\cos\theta\cos\varphi, \\
y(\theta,\,\varphi) &= R_0\sin\theta\cos\varphi, \\
z(\theta,\,\varphi) &= R_0p(\cos \varphi) \sin\varphi,
\end{aligned}
\end{equation}
where $\theta\in[-\pi,\,\pi)$, $\varphi\in[-\pi/2,\,\pi/2]$, $R_0=3.91\um$, and
$p(r)=0.105+r^2-0.56r^4$. These tests require a timestep of $k=0.1\us$ for stability.
With $\shear=1000\si{\per\second}$, an IB point requires at least 32 timesteps to transit
a grid cell, so unlike the tests using randomly placed IB points above, there will be
considerably more redundant computation. We first validate the fluid solver with the
elastic sphere before performing scaling tests, similar to those above, with RBCs.

\input{convergence-study}
\input{strong-scaling}
\input{weak-scaling}

% vim: cc=90 tw=89

\section{Summary} \label{sec:summary}

We presented {\revb Algorithm~\ref{algo:par-interp} for interpolation in parallel and introduced
a novel parallel algorithm, Algorithm~\ref{algo:par-spread},} for spreading forces from
time-dependent interfaces in the immersed boundary method. These algorithms have a
runtime that is independent of the Eulerian grid. This makes the parallelization useful
for cases where IB points inhabit only a small percentage of the Eulerian grid cells. We
also introduced two variants for spread which trade off dependence on the Eulerian grid,
in the form of a few vector adds and memory allocation, for improved runtimes for small
enough grids.

These algorithms exhibit nearly ideal scaling, on both the CPU and GPU, for problems of
fixed size and increasing number of threads, as well as for problems of increasing size
and number of threads.  We observed that on the GPU, larger problem sizes led to higher
speedup plateaus, indicating that larger problems on more capable hardware will achieve
even larger speedups than presented here.

{\revaa It is instructive to compare Algorithm~\ref{algo:par-spread}, which computes one value of
$\Dirac_h$ for each Lagrangian point per thread synchronization, with the parallelization
of~[McQueen and Peskin, 1997], which computes all values for $\Dirac_h$ for a subset of the
Lagrangian points each thread synchronization. Computing all values of $\Dirac_h$ at the
same time allows for minimizing calls to $\kernel$, while processing all Lagrangian
points simultaneously eliminates dependence on the background grid. It is straightforward
to construct a pathological case for which McQueen and Peskin's scheme performs poorly:
if all of the Lagrangian points lie within the same $s\times s$ columns of Eulerian grid
cells, then a single thread processes all of the Lagrangian points, reducing the scheme
to a serial one. The fine-grained approach in Algorithm~\ref{algo:par-spread} lacks this
pathology, making it more robust to IB point configurations.

Though efficient load-balancing for Lagrangian work requires special techniques
([Givelberg and Yelick, 2006, Bhalla et~al., 2019]), the worst-case scenario for distributed memory
parallelizations has every Lagrangian object mapped to the same device
([Erickson, 2010]). For that reason, single-node results are still relevant.
Nevertheless, we anticipate that our algorithms will perform well in multi-node systems
as they do not rely on partitioning the Eulerian or Lagrangian points.} Because
information about the Eulerian grid is encoded by the functions $\key$ and $\#$, an
optimized distributed-memory algorithm may need to use a different $\key$ or $\#$
function on each node to account for Eulerian grid partitioning. As simulations become
more computationally intensive, such algorithms will become more necessary. To that end,
future work involves adapting our algorithms for multi-node and multi-GPU systems.

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\begin{acks}
The authors acknowledge useful discussions with Dr. Aaron Barrett.
\end{acks}

\begin{dci}
The authors have no conflicts of interest to declare.
\end{dci}

\begin{funding}
The author(s) disclosed receipt of the following financial support for the research,
authorship, and/or publication of this article: This work was supported by the National
Science Foundation [grant numbers DMS-1521748, DMS-176898, CISE CCF-1714844].
\end{funding}

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