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Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory
In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and
operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the
Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and
then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a
certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér
Interpolation Problem for matrix rational functions. We then extend the
eBook
Practical linear algebra : a geometry toolbox
\"Practical Linear Algebra covers all the concepts in a traditional undergraduate-level linear algebra course, but with a focus on practical applications. The book develops these fundamental concepts in 2D and 3D with a strong emphasis on geometric understanding before presenting the general (n-dimensional) concept. The book does not employ a theorem/proof structure, and it spends very little time on tedious, by-hand calculations (e.g., reduction to row-echelon form), which in most job applications are performed by products such as Mathematica. Instead the book presents concepts through examples and applications. \"-- Provided by publisher.
Book
Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs
Consider a general linear Hamiltonian system
eBook
Spectral expansions of non-self-adjoint generalized Laguerre semigroups
We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local
Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a
subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the
framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between
this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As
a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our
methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to
equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert
space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are
led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these
norms.
eBook
Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations
In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous
reaction-diffusion equations:
The characterizations of these sets involve two new notions of generalized principal eigenvalues
for linear parabolic operators in unbounded domains. In particular, it allows us to show that
eBook
Recent trends in operator theory and applications: Workshop on Recent Trends in Operator Theory and Applications, May 3-5, 2018, the University of Memphis, Memphis, Tennessee
This volume contains the proceedings of the workshop on Recent Trends in Operator Theory and Applications (RTOTA 2018), held from May 3-5, 2018, at the University of Memphis, Memphis, Tennessee.The articles introduce topics from operator theory to graduate students and early career researchers. Each such article provides insightful references, selection of results with articulation to modern research and recent advances in the area.Topics addressed in this volume include: generalized numerical ranges and their application to study perturbation of operators, and connections to quantum error correction; a survey of results on Toeplitz operators, and applications of Toeplitz operators to the study of reproducing kernel functions; results on the 2-local reflexivity problem of a set of operators; topics from the theory of preservers; and recent trends on the study of quotients of tensor product spaces and tensor operators. It also includes research articles that present overviews of state-of-the-art techniques from operator theory as well as applications to recent research trends and open questions. A goal of all articles is to introduce topics within operator theory to the general public.
eBook
Weighted shifts on directed trees
A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated.
Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency
operators, on the other; they are called weighted shifts on directed trees. The basic properties of such operators, including
closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees
are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality,
cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights.
Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are
intensively studied mostly in the context of subnormality and complete hyperexpansivity of weighted shifts on them. A strict connection
of the latter with
eBook
Numerical Radius Inequalities for Finite Sums of Operators
A numerical radius inequality due to Kittaneh states that if
M
is a bounded linear operator on a complex separable Hilbert space, then
w
2
(
M
)
≤
1
2
M
∗
M
+
M
M
∗
.
We prove a numerical radius inequality for finite sums of operators, which is a considerable generalization of Kittaneh inequality: Let
M
1
,
M
2
,
…
,
M
n
,
N
1
,
N
2
,
…
,
N
n
be operators on a complex separable Hilbert space and let
f
,
g
be nonnegative continuous functions on
0
,
∞
which are satisfying the relation
f
(
a
)
g
(
a
)
=
a
for all
a
∈
0
,
∞
.
Then for
r
≥
2
,
w
r
∑
i
=
1
n
(
M
i
+
N
i
)
≤
2
r
-
2
Z
,
where
Z
=
∑
i
=
1
n
f
2
r
M
i
+
f
2
r
N
i
+
g
2
r
M
i
∗
+
g
2
r
N
i
∗
.
This inequality is attractive generalization of several recent inequalities.
Journal Article
Recent Trends in Operator Theory and Applications
This volume contains the proceedings of the workshop on Recent Trends in Operator Theory and Applications (RTOTA 2018), held from May 3-5, 2018, at the University of Memphis, Memphis, Tennessee. The articles introduce topics from operator theory to graduate students and early career researchers. Each such article provides insightful references, selection of results with articulation to modern research and recent advances in the area. Topics addressed in this volume include: generalized numerical ranges and their application to study perturbation of operators, and connections to quantum error correction; a survey of results on Toeplitz operators, and applications of Toeplitz operators to the study of reproducing kernel functions; results on the 2-local reflexivity problem of a set of operators; topics from the theory of preservers; and recent trends on the study of quotients of tensor product spaces and tensor operators. It also includes research articles that present overviews of state-of-the-art techniques from operator theory as well as applications to recent research trends and open questions. A goal of all articles is to introduce topics within operator theory to the general public.
eBook