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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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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.

A multidimensional extension of Bailey's transform is utilised to deduce two new generating relations of quite a general character. These expressions are then specialised to give more practical formulae in terms of Karlsson's generalised Kampé de Fériet functions which embody very many generating relations. A number of interesting special cases are given in an appendix which includes results involving Lauricella polynomials, generalised hypergeometric polynomials and the polynomials of Meixner, Charlier and Laguerre.

We consider singular integral operators of the form (a)$Z$_{1}L^{−1}Z_{2}, (b)$Z$_{1}Z_{2}L^{−1}, and (c)$L$^{−1}Z_{1}Z_{2}, where$Z$_{1}and$Z$_{2}are nonzero right-invariant vector fields, and$L$is the$L$^{2}-closure of a canonical Laplacian. The operators (a) are shown to be bounded on$L$^{p}for all$p$∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type ($p, p$) for any$p$∈[1, ∞).

It is shown that the Bargmann-Fock spaces of entire functions, A^{p}(C),$p$≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces.