In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the$C$^{*}-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces$A$^{2}(Ω) for a wide class of plane domains Ω⊂$C$, and in Fock spaces$A$^{2}($C$^{N}),$N$≧1.

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We determine the smallest Schatten class containing all integral operators with kernels in$L$_{p}(L_{p', q})^{symm}, where 2 <$p$∞ and 1≦$q$≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].

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No abstract uploaded!