Saichi Izumino

A Grüss-type inequality and its applications

We give a Grüss-type inequality which is a refinement of a result due to Dragomir and Agarwal. We also give its applications for the moments of random variables, guessing mappings, and Ozekis inequality.

Weighted geometric means of positive operators

A weighted version of the geometric mean of k () positive invertible operators is given. For operators and for nonnegative numbers such that , we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to if commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

Positive symmetric quotients and their selfadjoint extensions

We define a quotient B/A of bounded operators A and B on a Hilbert space H with a kernel condition kerA ⊂ kerB as the mapping Au→ Bu, u ∈ H. A quotient B/A is said to be positive symmetric if A∗B = B∗A ≥ 0. In this paper, we give a simple construction of positive selfadjoint extensions of a given positive symmetric quotient B/A.