Degree Calculations — The Hilbert Space Case

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.

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On the Operator Identity ∑ AkXBk ≡ 0

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-080-x ◽

1978 ◽

Vol 31 (4) ◽

pp. 845-857 ◽

Cited By ~ 37

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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Surjectivity of linear operators from a Banach space into itself

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700039538 ◽

2007 ◽

Vol 76 (1) ◽

pp. 143-154 ◽

Cited By ~ 3

Author(s):

Dimosthenis Drivaliaris ◽

Nikos Yannakakis

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Differential Operator ◽

Partial Differential Operator ◽

Second Order ◽

Linear Operators ◽

Partial Differential ◽

Space Case

We show that linear operators from a Banach space into itself which satisfy some relaxed strong accretivity conditions are invertible. Moreover, we characterise a particular class of such operators in the Hilbert space case. By doing so we manage to answer a problem posed by B. Ricceri, concerning a linear second order partial differential operator.

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Some surjectivity results for operators of generalized monotone type via a topological degree

Carpathian Journal of Mathematics ◽

10.37193/cjm.2018.03.07 ◽

2018 ◽

Vol 34 (3) ◽

pp. 333-340

Author(s):

SUK-JOON HONG ◽

◽

IN-SOOK KIM ◽

Keyword(s):

Hilbert Space ◽

Banach Spaces ◽

Topological Degree ◽

Degree Theory ◽

Monotone Operators ◽

Reflexive Banach Spaces ◽

Space Case ◽

Monotone Type

We introduce a topological degree for a class of operators of generalized monotone type in reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we give some surjectivity results for operators of generalized monotone type in reflexive Banach spaces. In the Hilbert space case, this reduces to the celebrated Browder-Minty theorem for monotone operators.

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

Complex Analysis and Operator Theory ◽

10.1007/s11785-020-01073-4 ◽

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.

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