Inverse problem in the theory of characteristic operator functions of unbounded operator bundles in a ?? space

1979 ◽  
Vol 31 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Yu. M. Arlinskii ◽  
V. A. Derkach
Keyword(s):  
Inverse Problem ◽  
Unbounded Operator ◽  
Characteristic Operator ◽  
Operator Functions

scholarly journals Correct solvability and representation of the solutions of Volterra integro-differential equations with exponential-fractional kernels

2019 ◽  
Vol 488 (5) ◽  
pp. 476-480
Author(s):  
V. V. Vlasov ◽  
N. A. Rautian
Keyword(s):  
Hilbert Space ◽  
Spectral Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Unbounded Operator ◽  
Strong Solutions ◽  
Abstract Form ◽  
Partial Differential ◽  
Operator Functions ◽  
Well Posed

For abstract integro-differential equations with unbounded operator coefficients in a Hilbert space, we study the well-posed solvability of initial problems and carry out spectral analysis of the operator functions that are symbols of these equations. This allows us to represent the strong solutions of these equations as series in exponentials corresponding to points of the spectrum of operator functions. The equations under study are the abstract form of linear integro-partial differential equations arising in viscoelasticity and several other important applications.

About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

2019 ◽  
pp. 211-217 ◽  
Author(s):  
Vasiliy. I Fomin
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
General Solution ◽  
Operator Equation ◽  
Bounded Operator ◽  
Characteristic Operator ◽  
Homogeneous Differential Equation ◽  
Operator Functions ◽  
Linear Homogeneous Differential Equation ◽  
Real Argument

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied

Unitary Systems and Characteristic Operator Functions

1993 ◽  
pp. 700-786
Author(s):  
I. Gohberg ◽  
M. A. Kaashoek ◽  
S. Goldberg
Keyword(s):  
Characteristic Operator ◽  
Operator Functions

scholarly journals Investigation of Operator Models Arising in Viscoelasticity Theory

2018 ◽  
Vol 64 (1) ◽  
pp. 60-73
Author(s):  
V V Vlasov ◽  
N A Rautian
Keyword(s):  
Hilbert Space ◽  
Spectral Analysis ◽  
Unbounded Operator ◽  
Integrodifferential Equations ◽  
Abstract Form ◽  
Partial Derivatives ◽  
Correct Solvability ◽  
Operator Functions

We study the correct solvability of initial problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. We do spectral analysis of operator-functions that are symbols of such equations. The equations under consideration are an abstract form of linear integrodifferential equations with partial derivatives arising in viscoelasticity theory and having a number of other important applications. We describe localization and structure of the spectrum of operatorfunctions that are symbols of such equations.

scholarly journals An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3415-3425 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Unbounded Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Normal Operator ◽  
Condition Numbers ◽  
Unbounded Operators ◽  
Similarity Condition ◽  
Von Neumann ◽  
Operator Functions

Let H be a linear unbounded operator in a separable Hilbert space. It is assumed the resolvent of H is a compact operator and H ? H* is a Schatten - von Neumann operator. Various integro-differential operators satisfy these conditions. Under certain assumptions it is shown that H is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS

2014 ◽  
Vol 97 (3) ◽  
pp. 331-342 ◽  
Author(s):  
MICHAEL GIL’
Keyword(s):  
Hilbert Space ◽  
Condition Number ◽  
Unbounded Operator ◽  
Differential Operators ◽  
Normal Operator ◽  
Condition Numbers ◽  
Unbounded Operators ◽  
Similarity Condition ◽  
Operator Functions ◽  
Schmidt Operator

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

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