Asymmetric truncated Toeplitz operators equal to the zero operator

Asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These operators are natural generalizations of truncated Toeplitz operators. In this paper we describe symbols of asymmetric truncated Toeplitz operators equal to the zero operator.


Introduction
Let H 2 denote the Hardy space of the unit disk D = {z : |z| < 1}, that is, the space of functions analytic in D with square summable Maclaurin coefficients.
Using the boundary values, one can identify H 2 with a closed subspace of L 2 (∂D), the subspace of functions whose Fourier coefficients with negative indices vanish. The orthogonal projection P from L 2 (∂D) onto H 2 , called the Szegö projection, is given by Note that if f ∈ L 1 (∂D), then the above integral still defines a function P f analytic in D.
The classical Toeplitz operator T ϕ with symbol ϕ ∈ L 2 (∂D) is defined on H 2 by is a reproducing kernel for the model space K α , i.e., for each f ∈ K α and w ∈ D, f (w) = f, k α w ( ·, · being the usual integral inner product). Observe that k α w is a bounded function for every w ∈ D. It follows that the subspace K ∞ α = K α ∩ H ∞ is dense in K α . If α(w) = 0, then k α w = k w , where k w is the Szegö kernel given by k w (z) = (1 − wz) −1 .
The function α is said to have a nontangential limit at η ∈ ∂D if there exists α(η) such that α(z) tends to α(η) as z ∈ D tends to η nontangentially (with |z − η| ≤ C(1 − |z|) for some fixed C > 0). We say that α has an angular derivative in the sense of Carathéodory (an ADC) at η ∈ ∂D if both α and α ′ have nontangential limits at η and |α(η)| = 1 (for more details see [9, pp. 33-37]). P. R. Ahern and D. N. Clark proved in [1,2], that α has an ADC at η ∈ ∂D if and only if every f ∈ K α has a nontangential limit f (η) at η. If α has an ADC at η and w tends to η nontangentially, then the reproducing kernels k α w tend in norm to the function k α η ∈ K α given by (1.1) with η in place of w. Moreover, f (η) = f, k α η for all f ∈ K α . Let P α denote the orthogonal projection from L 2 (∂D) onto K α . Then Just like with the Szegö projection, P α f is a function analytic in D for all f ∈ L 1 (∂D).
A truncated Toeplitz operator with a symbol ϕ ∈ L 2 (∂D) is the operator A α ϕ defined on the model space K α by . Densely defined on bounded functions, the operator A α ϕ can be seen as a compression to K α of the classical Toeplitz operator T ϕ on H 2 .
The study of truncated Toeplitz operators as a class began in 2007 with D. Sarason's paper [13]. In spite of similar definitions, there are many differences between truncated Toeplitz operators and the classical ones. One of the first results from [13] states that, unlike in the classical case, a truncated Toeplitz operator is not uniquely determined by its symbol. More precisely, A α ϕ = 0 if and only if ϕ ∈ αH 2 + αH 2 ([13, Thm. 3.1]). As a consequence, unbounded symbols can produce bounded truncated Toeplitz operators. Moreover, there exist bounded truncated Toeplitz operators for which no bounded symbols exist (see [3]). For more interesting results see [6,9,10,11,12].
Recently, the authors in [4] and [5] introduced a generalization of truncated Toeplitz operators, the so-called asymmetric truncated Toeplitz operators. Let α, β be two inner functions and let ϕ ∈ L 2 (∂D). An asymmetric truncated Toeplitz operator A α,β ϕ is the operator from K α into K β given by : ϕ ∈ L 2 (∂D) and A α,β ϕ is bounded} and T (α) = T (α, α). The purpose of this paper is to describe when an operator from T (α, β) is equal to the zero operator. The description is given in terms of the symbol of the operator. This was done in [4] and [5] for the case when β divides α, that is, when α/β is an inner function. It was proved in [4] and [5] that A α,β ϕ = 0 if and only if ϕ ∈ αH 2 + βH 2 . Here we show that this is true for all inner functions α and β. We also give some examples of rank-one asymmetric truncated Toeplitz operators.

Main result
In this section we prove the following.
We start with a simple technical lemma.
Lemma 2.2. Let α, β be two arbitrary inner functions. If then both α and β have no zeros in D, or at least one of the functions α or β is a constant function.
By the maximum modulus principle, α is a constant function. Hence, the inclusion (2.1) implies that β has no zeros in D, or α is a constant function. But (2.1) is equivalent to K β ⊂ αH 2 , and, by the same reasoning, (2.1) also implies that α has no zeros in D, or β is a constant function. This completes the proof. Lemma 2.2 can be rephrased as follows. If α, β are two nonconstant inner functions and at least one of them has a zero in D, then the inclusion K α ⊂ βH 2 does not hold. This allows us to prove the following version of Theorem 2.1. Proposition 2.3. Let α, β be two nonconstant inner functions such that each of them has a zero in D and let A α,β ϕ : K α → K β be a bounded asymmetric truncated Toeplitz operator with ϕ ∈ L 2 (∂D). Then A α,β ϕ = 0 if and only if ϕ ∈ αH 2 + βH 2 .
To give a proof of Theorem 2.1 we use the so-called Crofoot transform. For any inner function α and w ∈ D, the Crofoot transform J α w is the multiplication operator given by The Crofoot transform J α w is a unitary operator from K α onto K αw , where Lemma 2.4. Let α be an inner function and w ∈ D. For every z ∈ D we have Proof. Fix w, z ∈ D. The reproducing kernel k αw z is given by

It is known that the map
carries T (α) onto T (α w ) (see [6]). A similar result is true for the asymmetric truncated Toeplitz operators.
Proposition 2.5. Let α, β be two inner functions. Let a, b ∈ D and let the functions α a , β b and the operators J α a : K α → K αa , J β b : K β → K β b be defined as in (2.5) and (2.4), respectively. If A is a bounded linear operator from K α into K β , then A belongs to T (α, β) if and only if Proof. Let A be a bounded linear operator from K α into K β . Assume first that A belongs to . For every f ∈ K ∞ αa and z ∈ D we have By (2.6), .
To prove the other implication assume that But (α a ) a = α and (β b ) b = β, and, by the first part of the proof, Hence, A ∈ T (α, β). An easy computation shows that φ satisfies (2.7).
Corollary 2.6. If ϕ is in L 2 (∂D), then there is a pair of functions χ ∈ K α , ψ ∈ K β , such that A α,β ϕ = A α,β χ+ψ . If χ, ψ is one such pair, then the most general such pair is of the form χ − ck α 0 , ψ + ck β 0 , with c a scalar. Proof. The proof is analogous to the proofs given in [13] and [4].