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Advances in Complex Analysis and Its Applications in Operator Theory, Functional and Noncommutative Analysis

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Prof. Dr. Danko R. Jocić

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Department of Mathematics, University of Belgrade, 105104 Belgrade, Serbia
Interests: transformers in operator theory and matrix analysis; operator and norm inequalities; Bloch and weighted Lipschitz spaces; Bergman spaces in model theory for operators

Special Issue Information

Dear Colleagues,

Complex analysis is a very prosperous and inseparable part of mathematical analysis. Its traditional focus in on functions with one or several complex variables, especially those that are holomorphic in their domains. Cauchy and other representation formulas for holomorphic functions give rise to functional calculus with one or several operators (as variables), leading to the development of Cauchy–Dunford, Nagy–Foias, Taylor and spectral functional calculus.

Fourier and Laplace transforms are not just an important topic in mathematical analysis and other branches of mathematics and engineering, but also play a significant role in some Banach spaces of analytic functions, including, amongst others, Hardy, Bergman and Paley—Wiener spaces.

This Special Issue intends to present recent results and developments on the Banach spaces of complex and vector-valued functions, Fourier and Laplace transforms and transformers, and elementary and Birman–Solomyak transformers, which are given by double operator integrals on the ideals of compact operators, with applications in operator and norm inequalities. Also, works on the progress of functional calculus with several operators, model operators and model spaces, and its applications, will be welcomed in this Special Issue.

Prof. Dr. Danko R. Jocić
Guest Editor

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Research

14 pages, 337 KiB

Open AccessArticle

Norm Estimates for Remainders of Noncommutative Taylor Approximations for Laplace Transformers Defined by Hyperaccretive Operators

by Danko R. Jocić

Mathematics 2024, 12(19), 2986; https://doi.org/10.3390/math12192986 - 25 Sep 2024

Viewed by 770

Abstract

Let

H

be a separable complex Hilbert space,

B (H)

the algebra of bounded linear operators on

H

μ

a finite Borel measure on

R_{+}

with the finite

(n + 1)

-th moment, [...] Read more.

Let

H

be a separable complex Hilbert space,

B (H)

the algebra of bounded linear operators on

H

μ

a finite Borel measure on

R_{+}

with the finite

(n + 1)

-th moment,

f (z) : = \int_{R_{+}} e^{- t z} d μ (t)

for all

ℜ z ⩾ 0, C_{Ψ} (H)

, and

| | \cdot {| |}_{Ψ}

the ideal of compact operators and the norm associated to a symmetrically norming function

Ψ,

respectively. If

A, D \in B (H)

are accretive, then the Laplace transformer on

B (H),

X \mapsto \int_{R_{+}} e^{- t A} X e^{- t D} d μ (t)

is well defined for any

X \in B (H)

as is the newly introduced Taylor remainder transformer

R_{n} (f; D, A) X : = f (A) X - \sum_{k = 0}^{n} \frac{1}{k!} \sum_{i = 0}^{k} {(- 1)}^{i} (\binom{k}{i}) A^{k - i} X D^{i} f^{(k)} (D) .

A, D^{*}

are also

(n + 1)

-accretive,

\sum_{k = 0}^{n + 1} {(- 1)}^{k} (\binom{n + 1}{k}) A^{n + 1 - k} X D^{k} \in C_{Ψ} (H)

and

| | \cdot {| |}_{Ψ}

is Q* norm, then

| | \cdot {| |}_{Ψ}

norm estimates for

{(\sum_{k = 0}^{n + 1} (\binom{n + 1}{k}) A^{k} A^{n + 1 - k})}^{\frac{1}{2}} R_{n} (f; D, A) X {(\sum_{k = 0}^{n + 1} (\binom{n + 1}{k}) D^{n + 1 - k} D^{* k})}^{\frac{1}{2}}

are obtained as the spacial cases of the presented estimates for (also newly introduced) Taylor remainder transformers related to a pair of Laplace transformers, defined by a subclass of accretive operators. Full article

(This article belongs to the Special Issue Advances in Complex Analysis and Its Applications in Operator Theory, Functional and Noncommutative Analysis)

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