Pick’s theorems for dissipative operators

Let H be a complex Hilbert space and let A be a bounded linear transformation on H. For a complex-valued function f, which is analytic in a domain D of the complex plane containing the spectrum of A, let f (A) denote the operator on H defined by means of the Riesz-Dunford integral. In the present paper, several (presumably new) versions of Pick?s theorems are proved for f (A), where A is a dissipative operator (or a proper contraction) and f is a suitable analytic function in the domain D.

On the surjectivity of linear transformations

LetBbe a reflexive Banach space,Xa locally convex space andT:B→X(not necessarily bounded) linear transformation. A necessary and sufficient condition is obtained so that for a givenv∈Xthere is a solution for the equationTu=v. This result is used to discuss the existence of anLp-weak solution ofDu=vwhereDis a differential operator with smooth coefficients andv∈Lp.

Co-Rank of a Composition Operator

A composition operator CT on L2(X, Σ,m) is a bounded linear transformation induced by a mapping T : X → X via CTf = f∘ T.If m has no atoms then the co-rank of CT (i.e., dim is either zero or infinite. As a corollary, when m has no atoms, CT is a Fredholm operator iff it is invertible.

Composition operators on a functional Hilbert space

Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.

Boundary value problems for linear systems

SynopsisSuppose H and K are Hilbert spaces and H′0, H′ are closed subspaces of H so that H′0 ⊂ H′. Denote by P the orthogonal projection of H onto H′0, denote by g an element of k and by C a bounded linear transformation from H to K so that CC* = I, the identity on K. Denote CPC* by M. Given w in H′ one has the problem of finding u in H′ so thatThere are given conditions on M (or certain operators related to M) which imply convergence of a certain iteratively generated sequence to a solution to this problem. The equation Cu = g represents an inhomogeneous system of linear differential equations (ordinary, partial or functional) and the condition P(u − w) = u − w is an abstract representation of inhomogeneous boundary conditions for u.

On operators of class (N, k)

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.