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We obtain new and simple characterizations for the boundedness and compactness of weighted composition operators on the Fock space over C. We also describe all weighted composition operators that are normal or isometric.

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a b s t r a c tWe study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form (1 − ⟨z, w⟩) −γ for γ > 0. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.

Abstract. We obtain simple characterizations of unilateral and bilateral weighted shift operators that are m -isometric. We show that any such operator is a Hadamard product of 2 -isometries and 3 -isometries. We also study weighted shift operators whose powers are m -isometric.Mathematics subject classification (2010): 47B37, 47A65.

Abstract. For any function f in L ∞ (D), let T f denote the corresponding Toeplitz operator the Bergman space A 2 (D). A recent result of D. Luecking shows that if T f has finite rank then f must be the zero function. Using a refined version of this result, we show that if all except possibly one of the functions f1, . . . , fm are radial and T f 1 · · · T fm has finite rank, then one of these functions must be zero.

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For any α > −1, let A 2 α be the weighted Bergman space on the unit ball corresponding to the weight (1 − |z| 2 ) α . We show that if all except possibly one of the Toeplitz operators T f 1 , . . . , T fr are diagonal with respect to the standard orthonormal basis of A 2 α and T f 1 · · · T fr has finite rank, then one of the functions f1, . . . , fr must be the zero function. (2000). 47B35. Mathematics Subject Classification