scholarly journals Unitary Boundary Pairs for Isometric Operators in Pontryagin Spaces and Generalized Coresolvents

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
D. Baidiuk ◽  
V. Derkach ◽  
S. Hassi
Keyword(s):  
Hilbert Space ◽  
Schur Functions ◽  
Characteristic Functions ◽  
General Notion ◽  
Pontryagin Space ◽  
Isometric Operator ◽  
Pontryagin Spaces ◽  
Alternative Approach ◽  
Boundary Triple ◽  
Space Case

AbstractAn isometric operator V in a Pontryagin space $${{{\mathfrak {H}}}}$$ H is called standard, if its domain and the range are nondegenerate subspaces in $${{{\mathfrak {H}}}}$$ H . A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space case this notion generalizes the earlier concept of boundary triples for isometric operators and offers an alternative approach to study operator valued Schur functions without any additional invertibility requirements appearing in the ordinary boundary triple approach.

Characteristic Functions of Two-Operator Vessels in a Hilbert Space

1995 ◽  
pp. 187-233
Author(s):  
M. S. Livšic ◽  
N. Kravitsky ◽  
A. S. Markus ◽  
V. Vinnikov
Keyword(s):  
Hilbert Space ◽  
Characteristic Functions

scholarly journals Generalized Schur–Nevanlinna functions and their realizations

2020 ◽  
Vol 92 (5) ◽  
Author(s):  
Lassi Lilleberg
Keyword(s):  
Transfer Functions ◽  
Schur Functions ◽  
Inner Function ◽  
Pontryagin Space ◽  
State Spaces ◽  
Limit Values ◽  
Adjoint Systems ◽  
Generalized Schur Functions ◽  
Nevanlinna Functions ◽  
Weak Similarity

Abstract Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

On n-quasi-m-isometric operators

2016 ◽  
Vol 09 (04) ◽  
pp. 1650073 ◽  
Author(s):  
Salah Mecheri ◽  
T. Prasad
Keyword(s):  
Hilbert Space ◽  
Matrix Representation ◽  
Isometric Operator ◽  
Basic Properties

We introduce the class of [Formula: see text]-quasi-[Formula: see text]-isometric operators on Hilbert space. This generalizes the class of [Formula: see text]-isometric operators on Hilbert space introduced by Agler and Stankus. An operator [Formula: see text] is said to be [Formula: see text]-quasi-[Formula: see text]-isometric if [Formula: see text] In this paper [Formula: see text] matrix representation of a [Formula: see text]-quasi-[Formula: see text]-isometric operator is given. Using this representation we establish some basic properties of this class of operators.

VII.—On the Spectrum of Ordinary Second Order Differential Operators.

1969 ◽  
Vol 68 (2) ◽  
pp. 95-119 ◽  
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Analytic Function ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Space Theory ◽  
Alternative Approach

SynopsisThis paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.

scholarly journals Error Bound for Conic Inequality in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jiangxing Zhu ◽  
Qinghai He ◽  
Jinchuan Lin
Keyword(s):  
Hilbert Space ◽  
Error Bound ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Local Error ◽  
Local Error Bound ◽  
Local Error Bounds ◽  
Space Case ◽  
Vector Valued Functions ◽  
Vector Valued

We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.

Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

1997 ◽  
Author(s):  
Daniel Alpay ◽  
Aad Dijksma ◽  
James Rovnyak ◽  
Hendrik de Snoo
Keyword(s):  
Reproducing Kernel ◽  
Schur Functions ◽  
Pontryagin Spaces

On the Operator Identity ∑ AkXBk ≡ 0

1978 ◽  
Vol 31 (4) ◽  
pp. 845-857 ◽  
Author(s):  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Operator Identity ◽  
Image Position ◽  
Space Case ◽  
Necessary And Sufficient ◽  
Induced Mapping ◽  
Compact Map

Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by(1)We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the formFor example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

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