10.—An Abstract Relation for Multiparameter Eigenvalue Problems

1976 ◽  
Vol 74 ◽  
pp. 135-143 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Image Position ◽  
Symmetric Operators ◽  
Densely Defined

SynopsisConsider the multiparameter systemwhere ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equationwhere B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

scholarly journals Some notes on complex symmetric operators

2021 ◽  
Vol 2021 (1) ◽  
pp. 90-96
Author(s):  
Marcos S. Ferreira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Bounded Operator ◽  
Complex Symmetric ◽  
Symmetric Operators

Abstract In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * 𝒥T, where T is an unitary operator and 𝒥 f ( z ) = f ( z ¯ ) ¯ \mathcal{J}f\left( z \right) = \overline {f\left( {\bar z} \right)} with f ∈ H 2. Moreover we prove some relations of complex symmetry between the operators T and |T|, where T = U |T| is the polar decomposition of bounded operator T ∈ ℒ(ℋ) on the separable Hilbert space ℋ.

A Uniqueness Theorem on the Degenerate Cauchy Problem(1)

1975 ◽  
Vol 18 (3) ◽  
pp. 417-421 ◽  
Author(s):  
Chung-Lie Wang
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Uniqueness Theorem ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Continuous Linear ◽  
Linear Maps ◽  
Image Position ◽  
Degenerate Cauchy Problem ◽  
Densely Defined

In [4] Carroll and the author have treated the following problem(1)where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).

Existence conditions for eigenvalue problems generated by compact multiparameter operators

1984 ◽  
Vol 96 (3-4) ◽  
pp. 261-274 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Compact Resolvent ◽  
Multiparameter Eigenvalue Problem ◽  
Existence Conditions ◽  
Image Position ◽  
Adjoint Operators

SynopsisLet T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the conerelated to solubility of the multiparameter eigenvalue problemwith W(λ)−I nonpositive definite. The main result is as follows.Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

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

A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

1991 ◽  
Vol 110 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
C Algebra ◽  
Alternative Characterization ◽  
Image Position ◽  
Algebra Of Operators

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such thatas i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

On the Spectra of Unbounded Subnormal Operators

1986 ◽  
Vol 38 (5) ◽  
pp. 1135-1148 ◽  
Author(s):  
G. McDonald ◽  
C. Sundberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Line ◽  
Bounded Operator ◽  
Bounded Operators ◽  
The Real ◽  
Subnormal Operators ◽  
Hyponormal Operators ◽  
Image Position ◽  
Topological Boundary

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),

Variational Methods for One and Several Parameter Non-Linear Eigenvalue Problems

1981 ◽  
Vol 33 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Paul Binding
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Linear Operators ◽  
Main Thrust ◽  
Non Linear ◽  
Multiparameter Eigenvalue Problem ◽  
The One ◽  
Image Position ◽  
Parameter Problem

We shall consider a multiparameter eigenvalue problem of the form(1.1)where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

Application of g-frames in conjugate gradient

2016 ◽  
Vol 7 (3) ◽  
Author(s):  
Hassan Jamali ◽  
Neda Momeni
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Operator Equation ◽  
Conjugate Gradient ◽  
Separable Hilbert Space

AbstractThis paper proposes an iterative method for solving an operator equation on a separable Hilbert space

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