Fredholmness of multipliers on Hardy–Sobolev spaces

Spectral theory of multiplication operators on Hardy–Sobolev spaces

“…(N − β) − 2j + 1 < 0.It follows from(8) and the argument above again that I k,1 → 0 as k → ∞.If j = 1 and 0 < N − β = 1/2, thenI k,1 ≤ C D |Rψ(z)f k+1−N (z)| 2 dA 2(N −β)−1 (z) → 0, k → ∞, by dominated convergence. Finally, if j = 1 and N − β > 1/2 (which forces N ≥ 2), then 2(N − β) − 2j + 1 > 0, or 2(N − β) − 1 > 0.In this case, we recall from the remarks following Theorem 2.1 of[26] thatR N ψ ∈ A 2 2(N −β)−1 implies lim |z|→1 |R N ψ(z)|(1 − |z| 2 ) N −β+ 1 2 = 0,which is equivalent tolim |z|→1 |R N −1 ψ(z)|(1 − |z| 2 ) N −β− 1 2 = 0.…”

“…In this section we determine the spectrum of M ϕ : H 2 β → H 2 β for ϕ ∈ M β and β ∈ R. Note that the case of Hardy and Bergman spaces is well known; see [25] for example. Some preliminary work about the spectrum of multiplication operators on the Dirichlet space can be found in [8].…”

Unified Approach to Spectral Properties of Multipliers

“…Aside from obtaining a description of the spectrum for all spaces satisfying the mentioned properties, we also have to develop some new techniques to determine the essential spectrum of M u regarding the non-Hilbert space case. Other previous work regarding spectral and related properties of multiplication operators on analytic function spaces includes [3], [5], [8], [9], [14], [18] and [19].…”