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TitleOperator Theory in Harmonic and Non-commutative Analysis: 23rd International Workshop in Operator Theory and its Applications, Sydney, July 2012
Author
LanguageEnglish
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Total Pages260
Table of Contents
                            Contents
Introduction
Tent Spaces over Metric Measure Spaces under Doubling and Related Assumptions
	1. Introduction
	2. Spatial assumptions
	3. The basic tent space theory
		3.1. Initial definitions and consequences
		3.2. Duality, the vector-valued approach, and complex interpolation
			3.2.1. Midpoint results.
			3.2.2. Endpoint results.
		3.3. Change of aperture
		3.4. Relations between A and C
	Appendix: Assorted lemmas and notation
		A.1. Tents, cones, and shadows
		A.2. Measurability
		A.3. Interpolation
	Acknowledgements
	References
Remarks on Functional Calculus for Perturbed First-order Dirac Operators
	1. Introduction
	2. Abstract results
	3. First-order constant coefficients differential systems
	4. Perturbed first-order differential systems
	5. Relation to kernel/range decomposition
	6. Self-adjoint D and accretive B
	Acknowledgment
	References
(m,λ)-Berezin Transform and Approximation of Operators on Weighted Bergman Spaces over the Unit Ball
	1. Introduction
	2. Preliminaries
	3. The (m,λ)-Berezin transform
	4. Approximation by Toeplitz operators
	Appendix: Toeplitz operators with measure symbols
	References
Normal and Cohyponormal Weighted Composition Operators on H2
	1. Introduction
	2. Preliminaries
	3. Main results
	References
A Subnormal Toeplitz Completion Problem
	1. Hyponormality and subnormality of Toeplitz operators: A brief survey
		1.1. Which operators are subnormal ?
		1.2. (Block) Toeplitz operators and bounded type functions
		1.3. Hyponormality of Toeplitz operators
		1.4. Halmos’ Problem 5
		1.5. A special subnormal Toeplitz completion
	2. Subnormal Toeplitz completions
	Acknowledgment
	References
Generalized Repeated Interaction Model and Transfer Functions
	1. Introduction
	2. A generalised repeated interaction model
	3. Outgoing Cuntz scattering systems
	4. Λ-linear systems and transfer functions
	5. Transfer functions, observability and scattering
	6. Transfer functions and characteristic functions of liftings
	Acknowledgment
	References
Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy
	1. Introduction
	2. General spectral theory of T-1/2rT-1/2
	3. Distributional coefficients
	4. The case of periodic coefficients
	Appendix A. Relative boundedness and compactness of operators and forms
	Appendix B. Supersymmetric Dirac-type operators in a nutshell
	Appendix C. Sesquilinear forms and associated operators
	Acknowledgment
	References
Remarks on Spaces of Compact Operators between Reflexive Banach Spaces
	1. Introduction
	2. Injectivity of the canonical map between tensor products
	3. Isometric properties of the space of compact operators
	References
Harmonic Analysis and Stochastic Partial Differential Equations: The Stochastic Functional Calculus
	1. Introduction
	2. Multiple stochastic integrals
	3. Stochastic equations in Banach spaces
		3.1. Stochastic integration of vector-valued functions
		3.2. The van Neerven–Veraar–Weis approach to the stochastic integration of vector-valued functions
	4. The stochastic Dyson series in M-type 2 Banach spaces
		4.1. Sectorial operators
	5. Stochastic functional calculus
		5.1. H∞ functional calculus
		5.2. Random resolvents
		5.3. Further developments
	References
Subideals of Operators – A Survey and Introduction to Subideal-Traces
	1. Introduction
	2. Preliminaries
	3. Subideals of operators
	4. Comparison of subideals to B(H)-ideals
	5. Subideal-Traces
	References
Multipliers and Lp-operator Semigroups
	References
Taylor Approximations of Operator Functions
	1. Introduction
	2. Schatten class perturbations
		2.1. Spectral shift functions
		2.2. Proof strategy
		2.3. Operator Lipschitz functions
	3. Some natural generalizations
		3.1. Compact resolvents and similar conditions
		3.2. Operators in a semifinite von Neumann algebra
		3.3. General traces
	References
                        
Document Text Contents
Page 1

Operator Theory
Advances and Applications

Joseph A. Ball
Michael A. Dritschel
A.F.M. ter Elst
Pierre Portal
Denis Potapov
Editors

Operator Theory
in Harmonic and
Non-commutative
Analysis
23rd International Workshop in
Operator Theory and its Applications,
Sydney, July 2012

Page 130

Generalized Repeated Interaction Model and Transfer Functions 125


U,Ũ

is a multi-analytic operator ([15]) (also called analytic intertwining

operator in [3]) because


U,Ũ

(∑
α∈Λ̃

u(α)zαzj
)
= MΘ

U,Ũ

(∑
α∈Λ̃

u(α)zα
)

zj for j = 1, . . . , d,

i.e., MΘ
U,Ũ

intertwines with right translation. The noncommutative power series

ΘU,Ũ is called the symbol of MΘU,Ũ .

5. Transfer functions, observability and scattering

We would now establish that the transfer function can be derived from the coisom-
etry W of Section 2. In the last section d-tuple z = (z1, . . . , zd) of formal noncom-
muting indeterminates were employed. Treat (zα)α∈Λ̃ as an orthonormal basis of
�2(Λ̃,C). Assume Y and U to be the spaces associated with our model with uni-
taries U and Ũ as in the last section. It follows from Remark 3.7 that there exist
a unitary operator Γ̃ : (H̃ ⊗ K∞)◦ → �2(Λ̃,Y) defined by

Γ̃(V Cα y) := yz
ᾱ for all α ∈ Λ̃, y ∈ Y.

We observe the following intertwining relation:

Γ̃(V Cα y) = (Γ̃y)z
ᾱ. (5.1)

Similarly, using Theorem 3.6, we can define a unitary operator Γ : (H ⊗K∞)◦(=
(H◦ ⊕ G))→ H◦ ⊕ �2(Λ̃,U) by

Γ(̊h⊕ V Eα η) := h̊⊕ ηzᾱ for all α ∈ Λ̃
where h̊ ∈ H◦, η ∈ U . In this case the intertwining relation is

Γ(V Eα η) = (Γη)z
ᾱ. (5.2)

Using the coisometric operator W , which appears in Remark 2.3, we define ΓW
by the following commutative diagram:

(H⊗K∞)◦ W ��

Γ

��

(H̃ ⊗ K∞)◦

Γ̃
��

H◦ ⊕ �2(Λ̃,U) ΓW �� �2(Λ̃,Y),

(5.3)

i.e., ΓW = Γ̃WΓ
−1.

Theorem 5.1. ΓW defined by the above commutative diagram satisfies

ΓW |�2(Λ̃, U) = MΘU,Ũ .

Page 131

126 S. Dey and K.J. Haria

Proof. Using the intertwining relation V Cj W = W V
E
j from Remark 2.3, and equa-

tions (5.1) and (5.2) we obtain

ΓW (ηz
βzj) = Γ̃WΓ−1(ηzβzj) = Γ̃W V Ej V

E
β̄

η

= Γ̃V Cj V
C
β̄

W η = (Γ̃W η)zβzj = ΓW (ηz
β)zj

for η ∈ U , β ∈ Λ̃, j = 1, . . . , d. Hence, ΓW |�2(Λ̃, U) is a multi-analytic operator.
For computing its symbol we determine ΓW η for η ∈ U , where η is identified with
ηzφ ∈ �2(Λ̃,U). For α = αn−1 . . . α1 ∈ Λ̃ let Pα be the orthogonal projection onto

Γ̃−1{f ∈ �2(Λ̃,Y) : f = yzα for some y ∈ Y}
= V Cᾱ Y = Ũ∗1 . . . Ũ∗n−1(H̃ ⊗ �α1 ⊗ · · · ⊗ �αn−1 ⊗ (ΩKn )⊥ ⊗ ΩK[n+1,∞))

with Ũi’s as in Proposition 2.1.
Recall that the tuple E associated with the unitary U is a lifting of the tuple

C (associated with the unitary Ũ) and so E can be written as a block matrix in

terms of C as follows: Ej =

(
Cj 0
Bj Aj

)
for j = 1, . . . , d w.r.t. to the decomposition

H = H̃⊕H◦ where B and A are some row contractions. Because E is a coisometric
lifting of C we have

d∑
j=1

CjC

j = I and

d∑
j=1

CjB

j = 0

(cf. [5]). Now using these relations and equations (4.2), (4.3) and (4.4) it can be
easily verified that

PαŨ

1 . . . Ũ


nPnUn . . . U1η = PαŨ


1 . . . Ũ


mPmUm . . . U1η for all m ≥ n, η ∈ U .

Using the formula of W from Proposition 2.1 we obtain

PαW η = PαŨ

1 . . . Ũ


nPnUn . . . U1η for η ∈ U .

Finally for η ∈ U

PαŨ

1 . . . Ũ


nPnUn . . . U1η =

{
D̃η if n = 1, α = ∅,
V Cᾱ (C̃E


αn−1 . . . E


α2

F ∗α1η) if n = |α|+ 1 ≥ 2.
This implies for η ∈ U

Γ̃WΓ−1η = Γ̃W η = D̃η ⊕

|α|≥1

(C̃E∗αn−1 . . . E

α2

F ∗α1η)z
α.

Comparing this with equation (4.8) we conclude that ΓW |�2(Λ̃, U) = MΘU,Ũ . �
Note that the Theorem 4.1 and its proof concern the transfer function of the

Λ̃-linear system and has nothing to do with the scattering theory. Theorem 5.1,
on the other hand, is the scattering theory part in the sense of Lax–Phillips [12].
The same function MΘ

U,Ũ
relates the outgoing Fourier representation for a vector

in the ambient scattering Hilbert space to the incoming Fourier representation for

Page 259

Taylor Approximations of Operator Functions 255

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Anna Skripka
Department of Mathematics and Statistics
University of New Mexico
400 Yale Blvd NE, MSC01 1115
Albuquerque, NM 87131, USA
e-mail: [email protected]

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