A note on compact operators and operator matrices

Journal of Advances in Mathematics and Computer Science, 2022

The concept of a compact operator on a Hilbert space, H is an extension of the concept of a matrix acting on a finite-dimensional vector space. In Hilbert space, compact operators are precisely the closure of finite rank operators in the topology induced by the operator norm. In this paper, we provide an elementary exposition of compact linear operators in pre-Hilbert and Hilbert spaces. However, whenever advantageous, we may prove a few results in the general context of normed linear spaces. It is well known that strong convergence implies weak convergence but weak convergence does not imply strong convergence. We also show that an operator T ∈ B(H) is compact if and only if T maps every weakly convergent sequence in H to a strongly convergent sequence.

Linear Algebra and its Applications, 2013

We study the existence and characterization properties of compact Hermitian operators C on a separable Hilbert space H such that C ≤ C + D , for all D ∈ D(K(H) h) or equivalently C = min D∈D(K(H) h) C + D = dist C, D(K(H) h) where D(K(H) h) denotes the space of compact real diagonal operators in a fixed base of H and. is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal.

Commentationes Mathematicae Universitatis Carolinae, 2012

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Proceedings of the American Mathematical Society, 1975

If every function f f in the range of a bounded linear operator on L p {L_p} is equal to zero on a set of measure greater than a fixed number ϵ \epsilon , it is shown that there is a common set of measure ϵ \epsilon on which every function is zero. A decomposition theorem for such operators is proved.

Abstract and Applied Analysis, 2014

We obtain the necessary and sufficient conditions for an almost conservative matrix to define a compact operator. We also establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes(f,X), whereX=c,c0,l∞. These results are achieved by applying the Hausdorff measure of noncompactness.

2008

We construct a Hereditarily Indecomposable Banach space Xd with a Schauder basis (en)n∈N on which there exist strictly singular noncompact diagonal operators. Moreover, the space Ldiag(Xd) of diagonal operators with respect to the basis (en)n∈N contains an isomorphic copy of ℓ∞(N).

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. Let A,B 2 L(H) we define the generalized derivation A,B : L(H) 7! L(H) by A,B(X) = AX XB, we note A,A = A. If for all X 2 L(H) and for all T 2 ker A the inequality ||T (AX XA)|| || T||(*) holds, then we say that the range of A is orthogonal to kernel A in the sense of Birkho. The operator A 2 L(H) is said to be finite (17) if ||I (AX XA)|| 1(**) for all X 2 L(H), where I is the identity operator. The well-known Inequality (**) due to J.P.Williams (17) is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory (18)). This topic deals with minimizing the distance, measured by some norms or other, between a varying commutator XX X X and some fixed operator (12). In this paper we prove that a paranormal operator is finite and we present some generalized finite operators. An extension of Inequality (*) is...

JOURNAL OF OPERATOR THEORY

For every homogeneous Hilbertian operator space H, we construct a Hilbertian operator space X such that every infinite dimensional subquotient Y of X is completely indecomposable, and fails the Operator Approximation Property, yet H is completely finitely representable in Y. If H satisfies certain conditions, we also prove that every completely bounded map on such Y is a compact perturbation of a scalar.

Acta Scientiarum Mathematicarum

A theorem of P. A. Fillmore and J. P. Williams [Adv. Math. 7, 254-281 (1971; Zbl 0224.47009)] implies that a bounded operator A on a separable Hilbert space H is compact if and only if it satisfies lim n Ae n =0, or equivalently, lim n (Ae n |e n )=0 for each orthonormal basis (e n ) for H. In the present note this theorem is reproved using the fact observed by Halmos that each sequence of unit vectors weakly converging to 0 approximately contains an orthonormal subsequence. It is also noted that the stronger version of the theorem remains true, namely without the continuity assumption on A. In the second part of the note the bounded operators for which there exists an orthonormal basis such that either of the above equalities holds are exhibited and completely described.

1. Compact operators on Banach spaces 2. Appendix: total boundedness and Arzela-Ascoli treated compact operators as limiting cases of finite-rank operators.

Cumhuriyet Science Journal, 2020

Some bounded operators are part of this paper.Through this paper we shall obtain common properties of Some bounded operators in Г-Hilbert space. Also, introduced 2-self-adjoint operators and it's spectrum in Г-Hilbert Space. Characterizations of these operators are also part of this literature.

2009

In this work we will investigate how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases as an extension of the known method of matrix representation by ONBs. We will give basic definitions of the functions connecting infinite matrices defining bounded operators on l 2 and operators on H. We will show some structural results and give some examples. Furthermore in the case of Riesz bases we prove that those functions are isomorphisms. We are going to apply this idea to the connection of Hilbert-Schmidt operators and Frobenius matrices. Finally we will use this concept to show that every bounded operator is a generalized frame multiplier.

Integral Equations and Operator Theory, 1978

Acta Scientiarum Mathematicarum

Quaestiones Mathematicae, 2019

In this paper, we will consider matrices with entries in the space of operators B(H), where H is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in B(H). We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as H ∞ (D) and A(D) when dealing with upper triangular matrices.