Takayuki Furuta

A proof via operator means of an order preserving inequality

Abstract We give an alternative and simplified proof via operator means of an “order preserving inequality” on A and B in the case A ⩾ B ⩾ 0, where an “operator” is a bounded linear operator on a Hilbert space.

Extensions of the Fuglede-Putnam-type theorems to subnormal operators

At first we investigate the similarity between the Kleinecke-Shirokov theorem for subnormal operators and the Fuglede-Putnam theorem and also we show an asymptotic version of this similarity. These results generalize results of Ackermans, van Eijndhoven and Martens. Also we show two theorems on degree of approximation on subnormal derivation ranges. These results generalize results of Stampfli on degree of approximation on normal derivation ranges. The purpose of this paper is to show that the Fuglede-Putnam-type theorem on normal operators can certainly be generalized to subnormal operators.

The operator equation T(H1nT)n=K

Abstract Let H and K be bounded positive operators on a Hilbert space, and assume that H is nonsingular. This paper shows that if there exists a positive operator T such that T(H 1 n T) n =K for some natural number n , then, for any natural number m such that m ⩽ n there exists a positive operator T 1 such that T 1 (H 1 m T 1 ) m =K . In each case, there is at most one positive solution T and T 1 respectively.

A counterexample to a conjectured Hermitian matrix inequality

On donne un contre-exemple a une inegalite relative aux matrices hermitiennes conjecturee par N.N. Chan et Man Kam Kwong

A Hilbert-Schmidt norm inequality associated with the Fuglede-Putnam theorem

The familiar FugledemPutnam theorem asserts that AX=XB implies A X = XB* when A and B are normal. We prove that let A and B* be hyponormal operators and let C be hyponormal commuting with A* and also let D* be a hyponormal operator commuting with B respectively, then for every Hilbert--Schmidt operator X, the Hilbert--Schmidt norm of AXD+CXB is greater than or equal to the Hilbert--Schmidt norm of A*XD*+C*XB*. In particular, AXD=CXB implies A*XD*=C*XB*. If we strengthen the hyponormality conditions on A, B*, C and D* to quasinormality, we can relax Hilbert--Schmidt operator of the hypothesis on X to be every operator in B(H) and still retain the inequality under hypotheses that C commutes with A and satisfies an operator equation and also D* commutes withB* and satisfies another similar operator equation respectively.

Necessary and sufficient conditions for spectral sets

We shall show necessary and sufficient conditions for which closed set X in the complex plane is a spectral set of an operator T on a complex Hilbert space.