An extension of the method of the hypercircle to linear operator problems with unilateral constraints

1980 ◽  
Vol 85 (3-4) ◽  
pp. 173-193 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Boundary Value Problems ◽  
Best Approximation ◽  
Decomposition Theorem ◽  
Linear Operators ◽  
Unilateral Constraints ◽  
Abstract Boundary ◽  
Polar Cones ◽  
Approximation Problems

SynopsisBy using a Hilbert space decomposition theorem for two polar cones it is shown that the method of the hypercircle can be extended to determine solutions to best approximation problems involving unilateral constraints. The method is applied to abstract boundary value problems for linear operators involving such constraints.

Upper and lower bounds for solutions of linear operator problems with unilateral constraints

1977 ◽  
Vol 76 (2) ◽  
pp. 95-105 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Lower Bounds ◽  
Original Problem ◽  
Adjoint Operator ◽  
Decomposition Theorem ◽  
Positive Definite ◽  
Upper And Lower Bounds ◽  
Unilateral Constraints ◽  
Space Decomposition

SynopsisDual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.

A Note on the Adjoint of the Product of Operators

1970 ◽  
Vol 13 (1) ◽  
pp. 39-45
Author(s):  
Chia-Shiang Lin
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Closed Subspace ◽  
Linear Operators ◽  
Finite Codimension ◽  
Restricted Condition ◽  
Closed Linear Operators

Cordes and Labrousse ([2] p. 697), and Kaniel and Schechter ([6] p. 429) showed that if S and T are domain-dense closed linear operators on a Hilbert space H into itself, the range of S is closed in H and the codimension of the range of S is finite, then, (TS)* = S*T*. With a somewhat different approach and more restricted condition on S, the same assertion was obtained by Holland [5] recently, that S is a bounded everywhere-defined linear operator whose range is a closed subspace of finite codimension in H.

Representation of certain Linear Operators in Hilbert Space

1971 ◽  
Vol 23 (1) ◽  
pp. 132-150 ◽  
Author(s):  
Bernard Niel Harvey
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Separable Hilbert Space ◽  
Finite Dimension ◽  
Linear Operators ◽  
Inner Product ◽  
Indefinite Metric ◽  
Bounded Linear ◽  
Bilinear Functional

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

scholarly journals An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems

1988 ◽  
Vol 31 (1) ◽  
pp. 99-105 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Boundary Value ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Space Applications ◽  
Bounded Linear ◽  
Image Position

Let L(H) be the algebra of all bounded linear operators on a separable complex Hubert space H. In a recent paper [7], explicit expressions for solutions of a boundary value problem in the Hubert space H, of the typeare given in terms of solutions of an algebraic operator equation

scholarly journals Closed linear operators with domain containing their range

1984 ◽  
Vol 27 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Schôichi Ôta
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Unbounded Operators ◽  
Densely Defined ◽  
Closed Linear Operators

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

scholarly journals Weyl type theorems for selfadjoint operators on Krein spaces

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6001-6016
Author(s):  
Il An ◽  
Jaeseong Heo
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Krein Space ◽  
Linear Operators ◽  
Fredholm Theory ◽  
Bounded Linear ◽  
Selfadjoint Operators ◽  
Krein Spaces ◽  
Weyl Type Theorems ◽  
Kreĭn Space

In this paper, we introduce a notion of the J-kernel of a bounded linear operator on a Krein space and study the J-Fredholm theory for Krein space operators. Using J-Fredholm theory, we discuss when (a-)J-Weyl?s theorem or (a-)J-Browder?s theorem holds for bounded linear operators on a Krein space instead of a Hilbert space.

scholarly journals A functional calculus for sesquihermitian operators on quaternionic Hilbert space

1976 ◽  
Vol 21 (1) ◽  
pp. 96-107
Author(s):  
N. C. Powers
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Distribution ◽  
Functional Calculus ◽  
Complex Hilbert Space ◽  
Linear Mapping ◽  
Linear Operators ◽  
Commuting Operators ◽  
Real Linear ◽  
Quaternionic Hilbert Space

AbstractA continuous real-linear operator A = A0 + i1A1 + i2A2 + i3A3 on a quaternionic Hilbert space is called sesquihermitian if the linear operators Av are Hermitian; this condition is independent of the choice of quaternion basis (i1,i2,i3). The joint spectral distribution of the Av provides a functional calculus for sesquihermitian operators and real-valued C∞-functions on . This calculus is independent of the quaternion basis and extends naturally to quaternion-valued functions to give a continuous quaternion-linear mapping from the algebra of these functions to that of sesquihermitian operators. The mapping is not, in general, multiplicative unless the Av commute, in which case it agrees with that for several commuting operators on complex Hilbert space.

scholarly journals Boundary value problems for second-order partial differential equations with operator coefficients

2001 ◽  
Vol 6 (5) ◽  
pp. 253-266
Author(s):  
Kudratillo S. Fayazov ◽  
Eberhard Schock
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Linear Operators ◽  
Dense Domain ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Well Posed ◽  
Stability Of The Solution

LetΩ Tbe some bounded simply connected region inℝ 2with∂ Ω T=Γ¯1∩Γ¯2. We seek a functionu(x,t)((x,t)∈Ω T)with values in a Hilbert spaceHwhich satisfies the equationALu(x,t)=Bu(x,t)+f(x,t,u,u t),(x,t)∈Ω T, whereA(x,t),B(x,t)are families of linear operators (possibly unbounded) with everywhere dense domainD(Ddoes not depend on(x,t)) inHandLu(x,t)=u tt+a 11u xx+a 1u t+a 2u x. The valuesu(x,t);∂u(x,t)/∂nare given inΓ 1. This problem is not in general well posed in the sense of Hadamard. We give theorems of uniqueness and stability of the solution of the above problem.

scholarly journals Certain types of linear operators on probabilistic Hilbert space

2015 ◽  
Vol 3 (2) ◽  
pp. 81
Author(s):  
Rana Al-Muttalibi ◽  
Radhi M.A Ali
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Real Hilbert Space ◽  
Bounded Operator ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Continuous Operators

<p>The purpose of this paper is to introduce some definitions, properties and basic results that show the relation between F-bounded of linear operator in probabilistic Hilbert space and bounded operator in norm. In the paper, we prove that the adjoint operator in probabilistic Hilbert space is bounded. The notion of the continuous operators in probabilistic Hilbert space and some basic results are given. In addition, we note that every operator in probabilistic real Hilbert space is a self-adjoint Operator.</p>

scholarly journals Pseudospectra in a non-Archimedean Banach space and essential pseudospectra in Eω

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3961-3976
Author(s):  
Aymen Ammar ◽  
Ameni Bouchekoua ◽  
Aref Jeribi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Completely Continuous ◽  
Completely Continuous Operator

In this work, we introduce and study the pseudospectra and the essential pseudospectra of linear operators in a non-Archimedean Banach space and in the non-Archimedean Hilbert space E?, respectively. In particular, we characterize these pseudospectra. Furthermore, inspired by T. Diagana and F. Ramaroson [12], we establish a relationship between the essential pseudospectrum of a closed linear operator and the essential pseudospectrum of this closed linear operator perturbed by completely continuous operator in the non-Archimedean Hilbert space E?.

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