operator T will be denoted by ker T and ran(T) respectively. For any operator T 2B(H), let jTj= (TT)1/2, and consider the following standard de?nitions: normal if TT = TT and T is hyponormal if jTj2 jTj2 (i.e., equivalently, if kTxk kTxkfor every x 2H). An operator T is said to be -paranormal iff kTxk2 kT2xkkxk, for all x 2H, or equivalently, Journal of Inequalities and Applications Hyponormal Toeplitz operators with polynomial symbols on weighted Bergman spaces In Sung Hwang 0 1 3 Jongrak Lee 0 1 3 Se Won Park 0 2 0 Sungkyunkwan University , Suwon 1 Department of Mathematics 2 Department of Mathematics, Shingyeong University , Hwaseong, 445-741 , Korea 3 Department of Mathematics Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied. Cl iWiT)) will, as usual, denote the closure of WiT). An operator S is said to be similar to an operator T in case there exists an invertible operator A such that 5 = A ~l TA. In this note, all the operators will relate to a Hilbert space H. We shall prove the following theorem. Theorem. Let N be a hyponormal operator. OPEN PROBLEMS IN TOEPLITZ OPERATOR THEORY 135 Problem 10. (1) Does there exist a Toeplitz operator that is polynomially hyponormal but not subnormal? (2) Is every polynomially hyponormal Toeplitz operator rationally hyponor-mal? (3) Is every Toeplitz operator a von-Neumann operator? Problem 11. Identify subsets S of L1(T) for which the spectrum Normal operator From Wikipedia, the free encyclopedia In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H > H that commutes with its hermitian adjoint N*, that is: NN* = N*N. Normal operators are important because the spectral theorem holds for them. Spectral analysis of linear operators has always been one of the more active and important fields of operator theory, and of extensive interest to many operator theorists. Its devel­ opments usually are closely related to certain important problems in contemporary mathematics and physics. In the The nicest operators on V are those that are diagonalizable with respect to an orthonormal basis of V. These are the operators such that there is an orthonormal basis consisting of eigenvectors of V. The spectral theorem for complex inner product spaces shows that these are precisely the normal operators. Theorem 5 (Spectral Theorem). CONTINUITY AND HYPONORMAL OPERATORS 323 A converse of the above assertions concerning absolute continuity also holds, namely: (*) Every bounded absolutely continuous selfadoint operator A i__s th__e real part of some completely hyponormal operator. (**) Every absolutely continuous unitary operator U is the unitary factor in the polar factorization (1.3) of some completely hyponormal operator. is hyponormal.,E( ) is not empty. By Lemma 4.3, if S, is hyponormal, then there exists a function k in H 1with kkk 1 6 1 such that k - 2H . By the Cowen Theorem ([Cow]), this implies that T is hyponormal. Takanori Yamamoto (T. YAMAMOTO) Hyponormal Singular Integral Operators Hyponormal and Subnormal Toeplitz Operators Carl C. Cowen This paper is my view of the past, present, and future of Problem 5 of Halmos's 1970 lectures Ten Problems in Hilbert Space " [12] (see also [13]): Is every subnorma