SPECTRA OF TENSOR PRODUCTS AND ELEMENTARY OPERATORS

Let A,B be bounded operators on complex Banach spaces X and Y respectively. The tensor product X (?)_α Y denotes the completion of X(?)Y with respect to a quasi-uniform reasonable norm α. In the case X and Y are Hilbert spaces, Brown and Percy showed that σ(A(?)B)-σ(A)(?)σ(B). This work was generalized by Scheter and Dash, and it was further generalized by R. Harte. He computed the joint spectrum for certain system of elements in a tensor product of Banach algebras.

ON THE CHARACTER OF COMPACT OPERATOR SEQUENCE

Let E be a Banach space. If {T_m} is a sequence of uniform bounded operators such that {T_nx_n} is compact for any bounded sequence {x_n}, we say that {T_n} is a general collective cempact operator sequence.

AN INEQUALITY OF SEMI-HYPONORMAL OPERATORS

<正> Let T be an operator on the Hilbert space H with polar decomposition T = UP. If T satisfies P— UPU~* = D ≥0,

ON THE PUTNAM-FUGLEDE THEOREM

In [1—3], we have discussed problems on the Putnam-Fuglede theorem of non-normal operators which reduce AX=XB’（AXB=X） to A*X =XB* （A*XB* =X）. For the normal operators the following problems are considered: Let （N1, …,Nn） and （M1,…, Mn） be two groups of commuting normal operators, and we