Spectral theory of multiplication operators on Hardy–Sobolev spaces

Abstract

For a pointwise multiplier ϕ of the Hardy-Sobolev space H 2β on the open unit ball B n in C n , we study spectral properties of the multiplication operator M ϕ : H 2 β → H 2 β . In particular, we compute the spectrum and essential spectrum of M ϕ and develop the Fredholm theory for these operators.CAO:

Unified Approach to Spectral Properties of Multipliers

“…which can be found in Corollary 5 in [10]. The proof of the formula uses the derivative D, but the formula remains valid for all linear operators S that admit the law S(f g) = f Sg + gSf, f, g ∈ H(D), and for which the formula is valid for N = 1.…”

“…In a very recent article [10], Cao, He, and Zhu considered the multiplication operator M u acting on the Hardy-Sobolev Hilbert space and characterized the spectrum and essential spectrum of M u . In the present work, we extend and generalize the results obtained there from Hardy-Sobolev Hilbert space to the Bergman-Sobolev and Bloch-type spaces of the open unit ball B n of C n and weighted Hardy spaces of the open unit disk D with Muckenhoupt weights.…”

Fredholm composition operators and proper holomorphic mappings

“…The result above is well known to experts, but we are unable to locate a precise reference. It was stated as [, Lemma 8], but the reference there was not accurate. So we presented a proof here for the sake of completeness.…”

“…It is well known that the resulting space is independent of the choice of . These are called weighted Bergman spaces in and they have been studied in several other papers, including .…”

“…is the radial derivative of f . It is clear that R : H(B n ) → H(B n ) is a linear map and R k is well defined for any positive integer k. It is well known that the resulting space A 2 α is independent of the choice of k. These are called weighted Bergman spaces in [24] and they have been studied in several other papers, including [5].…”