Morisuke Hasumi

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ∞) such thatu−1 is operator monotone. We then look for another functionv such thatv(u−1) is operator monotone, namely,u(A)≦u(B) impliesv(A)≦v(B) for self-adjoint operatorsA andB.

Toeplitz Operators Acting on Generalized Functions

There have been extensive studies of Toeplitz operators mainly for Hardy classes and also for Bergman classes. In the present paper we will discuss some preliminary results concerning Toeplitz operators for Bergman classes. More explicitly, we look at the space of boundary functions of elements in the standard Bergman class B 2(D), consisting of functions analytic on the open unit disk D and square-summable with respect to the area measure. As the functions in B 2 (D) usually have boundary values only in the sense of Schwartz distributions, so we are led to the notion of Toeplitz operators acting on distributions.