Let T be a bounded, linear operator on a complex, separable, infinite dimensional Hilbert space H. We assume that T is an essentially isometric (resp. normal) operator, that is, IH-T*T (resp. TT* -T*T) is compact. For the compactness of S from the COMMUTANT of T, some necessary and sufficient conditions are found on S. Some related problems are also discussed.

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For a subset E of the bidisc D2, M={fÎH2(D2) :f=0on E} and N is the orthogonal complement of M in H2(D2) where H2(D2) is the two variable Hardy space on D2. We describe the finite rank COMMUTANTs of the restricted shifts Sz and Sw on N when E satisfies some natural condition. Moreover we give a sufficient condition for that the Pick interpolation is possible.

Given an inclusion NÌM of II1 factors with trivial relative COMMUTANT, this paper lists all operators x, yÎM such that the left N-module generated by x is equal to or contained in the right N-module generated by y.

Let B be the open unit ball in the n-space C-n and T-u be the Toeplitz operator on the Bergman space L-a(2)(B) with symbol u. In the present paper we:describe the essentially commuting Toeplitz operators on L-a(2)(B) with pluriharmonic symbols.

In this paper, we study coposinormal composition operators and posinormal weighted composition operators on the Hardy space H2(D). We show that if W,' is coposinormal on H2(D), then never vanishes on D also we prove that ' is univalent. Moreover, we study the COMMUTANT of a coposinormal weighted composition operator.

Let H be a complex Hilbert space and let {Bi} (-∞ +∞) be a uniformlybounded sequence of invertible operators on H. The invertible bilateral operator valued weighted shift B with weight sequence {Bi} is the operator on l2 (H ) = ...+H+H +H ... defined by B(..., x-1, x0 , x1,...) = (...,B-2 x-2 , B-1x-1, B0 x0 ,...) where x=(...,X-1,X0,X1,...)έ L2(H) , and the symbol "V" is used to denote the location of the zeroth coordinate. Our goal is to give sufficient conditions so that Alg Lat B be a subset of the COMMUTANT of B.

We will consider multiplication operators on Hilbert spaces of analyticfunctions on a domain $\omega \subset \cmark $. We determine the COMMUTANTs of certainmultiplication operators with adjoints in a Cowen-Douglas class operator.