New spaces of matrices with operator entries

Annales Academiae Scientiarum Fennicae Mathematica, 2012

We consider the action of the Hilbert matrix operator, H, on the Hardy space H 1 , weighted Hardy spaces H p α (α ≥ 0), Bergman spaces with logarithmic weights, etc. In particular, we extend Diamantopoulos-Siskakis result by proving that H maps H p α into H p α if and only if α+1/p < 1. A criterion for Hf to belong to H 1 is given provided the coefficients of f are nonnegative.

Houston Journal of Mathematics, 1986

Let G be a countable union of annuli centered at 0 and contained in the unit disc D; let 1G be the characteristic function of G. Let T G be the Toeplitz operator on the Bergman space A 2 with symbol 1G. We show that the essential spectrum of T G is connected, and we give upper and lower bounds on the spectrum and essential spectrum in terms of the radii of the annuli. 1. Introduction. Let C be the complex plane and let D be the open unit disc. Let dA be normalized (Lebesgue) area measure on D; define Løø(D) as the complex-valued functions defined on D which are measurable and essentially bounded with respect to dA. Define L2(D) to be the complex-valued functions defined on D which are measurable and square-integrable with respect to dA. L2(D) is a Hilbert _ space with inner product (f,g)L2 = fD f fg dA. Define the Bergman space A 2 to be the analytic functions in L2(D). It is well known that A 2 is a closed subspace of L2(D) and is thus a Hilbert space (see [4]). Let P be the projection from L2(D) onto A 2. Let fC Løø(D); the Toeplitz operator on A 2 with symbol f, denoted Tf, is defined by Tf(g) = P(fg), g C A 2. We shall denote the spectrum of Tf by o(Tf). Let B(A 2) be the Banach algebra of bounded linear operators on A2; let •c(A 2) denote the ideal of compact operators on A 2. The quotient algebra B(A2)/•c(A 2) is referred to as the Calkin algebra, and the quotient map rr: B(A 2)-• B(A2)fic(A 2) is referred to as the Calkin map. The essential spectrum of Tf, denoted oe(Tf), is the spectrum of rr(Tf) in the Calkin algebra. In this work we shall consider Toeplitz operators on A 2 for which the symbol is the characteristic (indicator) function of a measurable subset G of D. An important 397 398 JAMES W. LARK, III application of results concerning these operators is in the work of Voas [8]. Let 1G denote the characteristic function of G; i.e., 1G(Z) = 1 if z E G, 1G(Z) = 0 if z • G. We shall use the symbol T G to represent the operator T1G on A 2. It is easy to see that T G is self-adjoint, and that o(T G) _C [0,1]. Since T G is self-adjoint (and thus rr(TG)is self-adjoint), then sup o(T G) = sup{X: 3, E O(TG)} = I[TGII and sup oe(T G) = sup{X: 3, G oe(TG)} = I[TGlle (the essential norm of TG).

Mathematical Transactions of the Academy of Sciences of the Lithuanian SSR, 1973

HILBERT SPACES OF OPERATOR-VALUED FUNCTIONS E. Senkene and A, Tempel'man UDC 519,21 Hilbert spaces of scalar functions with reproducing kernels were introduced and studied in [1]. In [3] and [6] the properties and probability-theoretical applications of Hilbert spaces of vector-valued functions with operator-valued reproducing kernels were considered. The present paper is devoted to the study of Hilbert spaces of operator-valued functions with reproduction kernels; a number of facts which are well known in the scalar case (cf. [1]) are extended to such spaces. Let :~C be a separable Hilbert space; < .,. > :)~ and !!'ll~ denote the scalar product and norm in ~E; ~C is the Banaeh ring of all nuclear operators in }~ (with norm I[A ]]c}~=tr [(AA*)]~); • is the Banaeh algebra of linear operators (with norm It A tip = sup [I Ax [I iC); T is an arbitrary set; H is the Hilbert space con-. , ~II:E= i ' sisting of ~,C valued functions defined on T; <., * >H and I1 ' tl H denote the scalar product and norm in H. Definition 1. A ~C-valued function R(s, t), s, t, tE T, is called an operator-valued reproducing kernel of the space H if for any function A(.)~H and for any Be2 and s~ T a) the function BR(s, .)EH; b) <A(.), BR(s, ")>H = tr[A(s)B*], where tr[A(s)B*] denotes the trace of the operator A(s)B* (the "reproducing" property). We note that if B *= x | y is a one-dimensional operator defined by the formula (x| < z, y> }~x, then tr [A (s)B*]= < A (s)x, y> ??; thus, on the basis of the properties of the function A(.) as an element of the space H we can reconstruct its value at each point of s ~ T; in particular, if {e 1, e 2, e s .... } is a basis of }~, then a:~ (s)=~r [A (~) (e~| < 4 (s) e~, ej> }~ are the matrix elements of the operator A(s) with respect to this basis.

Oberwolfach Reports, 2000

The major topics discussed in this workshop were invariant subspaces of linear operators on Banach spaces of analytic functions, the ideal structure of H ∞ , asymptotics for condition numbers of large matrices, and questions related to composition operators, frequently hypercyclic operators, subnormal operators and generalized Cesàro operators. A list of open problems raised at this workshop is also included.

Journal of Functional Analysis

We study certain interpolation problems for analytic 2 × 2 matrix-valued functions on the unit disc. We obtain a new solvability criterion for one such problem, a special case of the µ-synthesis problem from robust control theory. For certain domains X in C 2 and C 3 we describe a rich structure of interconnections between four objects: the set of analytic functions from the disc into X , the 2 × 2 matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. This rich structure combines with the classical realisation formula and Hilbert space models in the sense of Agler to give an effective method for the construction of the required interpolating functions.

2021

As an extension to the study of Toeplitz operators on the Bergman space, the notion of H-Toeplitz operators Bφ is introduced and studied. Necessary and sufficient conditions under which H-Toeplitz operators become co-isometry and partial isometry are obtained. Some of the invariant subspaces and kernels of H-Toeplitz operators are studied. We have obtained the conditions for the compactness and Fredholmness for H-Toeplitz operators. In particular, it has been shown that a non-zero HToeplitz operator can not be a Fredholm operator on the Bergman space. Moreover, we have also discussed the necessary and sufficient conditions for commutativity of H-Toeplitz operators.

Mediterranean Journal of Mathematics, 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 (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vectorvalued functions.

Integral Equations and Operator Theory, 2009

We study the analogues of de Branges-Rovnyak spaces in the Banach space case. An important role is played by self-adjoint operators from the dual of a Banach space into the Banach space itself. A factorization theorem for such operators is proved in the case when they have a finite number of negative squares.

Rocky Mountain Journal of Mathematics, 2014