The essential norm of operators on the Bergman space of vector–valued functions on the unit ball

Abstract

Let A p α (B n ; C d ) be the weighted Bergman space on the unit ball B n of C n of functions taking values in C d . For 1 < p < ∞ let T p,α be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix-valued symbols (this is called the Toeplitz algebra in the case d = 1). We show that every S ∈ T p,α can be approximated by localized operators. This will be used to obtain several equivalent expressions for the essential norm of operators in T p,α . We then use this to characterize compact operators in A p α (B n ; C d ). The main result generalizes previous results and states that an operator in A p α (B n ; C d ) is compact if only if it is in T p,α and its Berezin transform vanishes on the boundary.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.

The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

“…A version of this result in the classical Bergman space setting was first proved by Suárez in [31]. Related results were later given in [4,20,22,25]. …”

A Reproducing Kernel Thesis for Operators on $\ell^2$--valued Bergman-type Function Spaces

“…A version of this result in the classical Bergman space setting was first proved by Suárez in [31]. Related results were later given in [4,20,22,25]. Proposition 4.3.…”