Model spaces and Toeplitz kernels in reflexive Hardy space

Abstract

Abstract. This paper considers model spaces in an H p setting. The existence of unbounded functions and the characterisation of maximal functions in a model space are studied, and decomposition results for Toeplitz kernels, in terms of model spaces, are established.Mathematics subject classification (2010): 47B35, 30H10.

Spectral properties of truncated Toeplitz operators by equivalence after extension

“…By doing so, we not only obtain various results that are new even for p = 2, but we also shed light on whether the properties that are studied, namely spectral properties of TTO, depend on the existence of an underlying Hilbert space structure, or on the value of p. In fact, properties such as Fredholmness, invertibility and the dimensions of the kernels and the cokernels of Toeplitz operators in the Hardy spaces H + p may depend on the value of p ∈ (1, ∞); it is easy to find examples of this behaviour by considering piecewise continuous symbols of the form g α (ξ) = ( ξ−i ξ+i ) α ( [12,21,23]. One would expect the same to hold for TTO defined in a model space K p θ := H + p ∩ θH − p , where θ is an inner function; however, somewhat surprisingly, the results obtained for the various classes of TTO considered in this paper do not depend on p. Note however, that in general the space K p θ on which the TTO are defined does depend on p: see, for example [8,14]. For example, this is the case for any infinite Blaschke product θ whose zeroes are not bounded away from the real axis.…”

“…are given by Dyakonov [13] (see also [14,15]) and some further equivalent conditions are given in [8]. Under these circumstances,θ 1 β 1 ker A θ g =θ 2 β ker(A θ g ) * .…”

“…It follows that we must have 8) and it is clear that a necessary and sufficient condition for the kernel of T G R (or, equivalently, A θ R ) to be nontrivial is that, for some polynomials Q 1 and Q 2 , with deg Q 1 < N + and deg Q 2 < N − , the conditions in (5.4) are satisfied.…”

Multipliers and equivalences between Toeplitz kernels

“…Various results regarding the dimension of ker Tz θh can also be found in [4] and [6]. Namely, if θ is a finite Blaschke product, ker Tz θh and ker Tz h are both finite dimensional or not and, for dim ker Tz h < ∞, we have dim ker Tz θh = max{0, dim ker Tz h − k}, where k is the degree of θ ([6] Theorem 6.2).…”

Toeplitz kernels and model spaces

“…[9]). If g ∈ L ∞ (T) and θ is a finite Blaschke product, then dim ker T g < ∞ if and only if dim ker T θg < ∞, and ker T g is finite-dimensional if and only if there exists k 0…”