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Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures.When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator Tμ with symbol μ, as an unbounded linear operator on the Hardy space. We then study various properties of Tμ. Among them, there is a theorem that the domain of Tμ is represented by a trichotomy. Also, it was shown that if the domain of Tμ contains at least one polynomial, then Tμ is densely defined. In addition, we give evidence for the conjecture that Tμ with a singular measure μ reduces to a trivial linear operator.


Introduction
A classical Toeplitz operator is the compression of a multiplication operator on the Lebesgue space L 2 ðT Þ of the unit circle T to the Hardy space H 2 ðT Þ. The study of Toeplitz operators seems to have originated from the paper of Toeplitz [1]. In the paper [2], he used Toeplitz matrices to characterize nonnegative continuous functions on the unit circle in terms of their Fourier coefficients. The remarkable paper of Brown and Halmos [3] started the systematic study of spectral properties of Toeplitz operators. Since then, the theory of Toeplitz operators has been studied in various ways. Recently, the theory of Toeplitz operators has been studied in a variety of settings and connections with other fields. One direction is to deal with Toeplitz operators on reproducing kernel spaces like Bergman spaces, Dirichlet spaces, or Fock spaces (cf. [4][5][6][7][8]). Another direction is to study Toeplitz operators with operator-valued symbols (cf. [9][10][11]). Also, truncated Toeplitz operators have attracted attention. A systematic approach on truncated Toeplitz operators can be found in the paper of Sarason in 2007 [12]. In that paper, he has used "compatible" measures to describe bounded truncated Toeplitz operators. The boundedness of infinite Hankel matrices is also related to the compatibility of measures: the infinite Hankel matrix of the moment of a nonnegative Carleson measure is bounded and vice versa [13]. (For related recent studies, see [14].) These works inspired us to consider Toeplitz operators whose symbols are measures. The Toeplitz operators whose symbols are measures have been studied in the setting of Bergman spaces and other spaces (cf. [15], chapter 7).
In this paper, we consider Toeplitz operators on the Hardy space, whose symbols are measures. In this study, unbounded Toeplitz operators arise naturally. When studying unbounded Toeplitz operators, it was usually considered that the symbols come from L 2 ðT Þ. In 2008, Sarason [16] treated not only the case of L 2 ðT Þ-symbols but the case of analytic functions on the open unit disk D. It is natural to attempt to extend the symbols of Toeplitz operators to measures, because the initial reasearch for them was related to the moment problem. As mentioned before, Toeplitz and Hankel operators associated with measures can be seen in the papers [13] and [12]. In this paper, we provide an explicit definition of Toeplitz operators whose symbols are complex Borel measures and then consider their unbounded operator theory. As the study on Toeplitz operators whose symbols are functions shows the interplay between function theory and operator theory, the study on Toeplitz operators whose symbols are measures is also expected to show the interplay between measure theory and operator theory.
Our consideration for the symbol of a Toeplitz operator, denoted by T μ , is a complex Borel measure μ on the unit cir-cle. When we study an unbounded linear operator, we usually assume that its domain is dense, i.e., the operator is densely defined. Hence, one may ask if T μ is densely defined, i.e., the domain is dense in H 2 . Toeplitz operators with L 2 -symbols are always densely defined. Unlike when the symbol is a function, it does not seem easy to answer the question. Nonetheless, we will show that the domain of T μ is represented by a trichotomy (Theorem 8). In particular, we can show that if the domain of T μ contains at least one polynomial, then T μ is densely defined (Proposition 10). We also give evidence for the conjecture that the cases of singular measures induce trivial linear operators (Theorem 15).
The organization of this paper is as follows. In Section 2, we give notations, definitions, and preliminary facts, which will be used in the sequel. In Section 3, we give a formal definition of Toeplitz operators whose symbols are complex Borel measures on T and then investigate their properties in the viewpoint of unbounded linear operator theory.

Preliminaries
Let T be the unit circle in the complex plane. Let m be the normalized Lebesgue measure on T , so that mðT Þ = 1. For 1 ≤ p ≤ ∞, we write L p ðT Þ = L p ðT , mÞ for the Lebesgue space on T and H p ðT Þ for the Hardy space on T . Note that H p ðT Þ is a closed subspace of L p ðT Þ.
Let D be the open unit disk and let D be the closed unit disk in the complex plane. Let C A ðDÞ denote the disk algebra, i.e., the set of all continuous functions on D which is analytic on D.
For 1 ≤ p ≤ ∞, we write H p ðDÞ for the Hardy space on D. Two spaces H p ðDÞ and H p ðT Þ are identified via nontangential limits and Poisson integral. Thus, we often write H p to denote the both of them. The norm in L p ðT Þ (or H p ðDÞ) will be denoted by ∥·∥ p and the inner product in L 2 ðT Þ (or H 2 ðDÞ) will be denoted by h·, · i. We refer the reader to the texts [17][18][19] and [20] for details of Hardy spaces.
The shift operator and its adjoint are one of the most interesting operators on the Hardy space. For convenience, we define them on HðDÞ, the class of all analytic functions on D. For f ∈ HðDÞ, define The operators S and S * are often called the unilateral shift and the backward shift, respectively. We refer the reader to the text [21] which treats the shift operator in great detail.
One of the most remarkable theorems in analysis is Beurling's theorem (cf. [18,20,22]), which characterizes all S-invariant subspaces of H 2 . (We use the term "subspace" for a closed linear subspace.) For a nonzero subspace M of H 2 , M is S-invariant if and only if for some inner function θ ∈ H ∞ . A bounded analytic function θ on D is called an inner function if its radial limit θ * ðe it Þ = lim r→1− θðre it Þ has a unit modulus for almost all e it ∈ T . If an inner function has no zero in D, we call it a singular inner function. Let MðT Þ be the set of all complex (finite) Borel measures on T . Note that MðT Þ is a Banach space with the total variation norm ∥μ∥ = | μ|ðT Þ, where |μ | is the total variation measure of μ. We may regard the normalized Lebesgue measure m as a finite positive Borel measure. Hence, m ∈ MðT Þ. We write B T for the σ-algebra of all Borel sets in T . We say μ is singular if μ⊥m.
For μ ∈ MðT Þ, the nth Fourier-Stieltjes coefficient of μ is given by For any μ ∈ MðT Þ, the bilateral sequence b μ = fb μðnÞg n∈ℤ is bounded and the mapping μ ↦ b μ is a bounded linear transformation from MðT Þ into ℓ ∞ ðℤÞ. Note that the mapping μ ↦ b μ is one-to-one, and hence, a measure μ ∈ MðT Þ is completely determined by its Fourier-Stieltjes coefficients. By the theorem of F. and M. Riesz, if μ ∈ MðT Þ is analytic, i.e., b μðnÞ = 0 for all n ≤ 0, then μ ≪ m and dμ/dm ∈ H 1 ðT Þ; in other words, μ = f · m for some f ∈ H 1 ðT Þ.
For the definition of Toeplitz operators whose symbols are measures, we use the Cauchy transform as the "projection" of measures. For this reason, we use the notation Pμ instead of Kμ for the Cauchy transform of μ. We refer the reader to the text [23] for thorough treatments of the Cauchy transform. For μ ∈ MðT Þ, the analytic function Pμ on D, given by is called the Cauchy transform of μ. Clearly, the mapping P is a linear transformation from MðT Þ into HðDÞ. We may regard f ∈ L 1 ðT Þ as the absolutely continuous measure f · m ∈ MðT Þ. Hence, we denote Pð f · mÞ by Pf , i.e., (Clearly, d f · mðnÞ =f ðnÞ.) As we have identified H 2 ðDÞ with H 2 ðT Þ, the mapping P may be regarded as the 2 Journal of Function Spaces orthogonal projection of L 2 ðT Þ onto H 2 ðT Þ (the so-called Riesz projection).
given by (Recall that every function in H 2 ðDÞ may be identified with its nontangential limit function which belongs to H 2 ðT Þ.) for every i, j ∈ ℕ ∪ f0g. Hence, the matrix representation of T φ with respect to the orthonormal basis f1, z, A matrix of this form is called a Toeplitz matrix; in other words, an infinite matrix fα i,j g i,j≥0 is called a Toeplitz matrix if For a bilateral sequence s = fs n g n∈ℤ of complex numbers, we denote by TðsÞ the infinite Toeplitz matrix corresponding to s, i.e., TðsÞ is the infinite matrix whose ði, jÞ-entry is s i−j . Note that if φ ∈ L 2 ðT Þ, then the matrix representations of T φ is Tðb φÞ. For n ∈ ℕ ∪ f0g, we denote by T n ðsÞ the ðn + 1Þ × ðn + 1Þ Toeplitz matrix corresponding to s, i.e.,

The Main Results
Let μ be a complex Borel measure on T . For any function f ∈ C A ðDÞ, f · μ is a complex Borel measure on T , and hence, the Cauchy transform Pð f · μÞ is an analytic function on D. Define It is easy to show that DðT μ Þ is a linear manifold of H 2 ðDÞ. Now define Then, T μ is a linear operator on H 2 ðDÞ with domain DðT μ Þ. Definition 1. The operator T μ is called the Toeplitz operator with symbol μ.
We begin with the following: Suppose that μ ≪ m and the Radon-Nikodym derivative φ = dμ/dm belongs to L 2 ðT Þ. Then, DðT μ Þ = C A ðDÞ and for every f ∈ C A ðDÞ.
Proposition 2 shows that the notion of T μ is a kind of generalization of the Toeplitz operators whose symbols are L 2 -functions.

Remark 3.
(a) Toeplitz operators with L 1 -symbols: every function φ ∈ L 1 ðT Þ would be regarded as the absolutely continuous measure φ · m ∈ MðT Þ. Hence, we may use Definition 1 to define Toeplitz operators with for f ∈ DðT μ Þ.
for f ∈ DðT μ Þ. This shows that a Toeplitz operator with H 1 -symbol behaves as a multiplication. Notice that the action of T μ is the same as that of T φ defined in ( [16], Section 5). (In that paper, the domain of T φ is given by such that a is an outer function, að0Þ > 0, and jaj 2 + jbj 2 = 1 on T . In this case, DðT φ Þ = aH 2 ðDÞ (cf. [16]). It follows that Since a is an outer function, it follows that aH 2 ðDÞ is dense in H 2 ðDÞ.
Question: is aH 2 ∩ C A ðDÞ dense in H 2 ?
We give some concrete examples.
How large is the domain DðT μ Þ? Suppose that g ∈ C A ðDÞ and gð1Þ ≠ 0. Then, there exists a constant c > 0 such that |g | ≥c on a neighborhood of ζ = 1. It follows that φg ∉ H 2 ðDÞ. Hence, g ∉ DðT μ Þ. This shows that On the other hand, if r > 0 and if ψ r is the function in C A ðDÞ which satisfies ðψ r ðzÞÞ 1/r = 1 − z and ψ r ð0Þ = 1, then, for every g ∈ C A ðDÞ, and hence, φψ r g ∈ H 2 ðDÞ, i.e., ψ r g ∈ DðT μ Þ. It follows that Since In particular, DðT μ Þ contains all polynomials vanishing at ζ = 1.
(c) The Cantor middle-third measure: let C denote the Cantor ternary set and let φ be the Cantor function, i.e., for Journal of Function Spaces and φðxÞ = sup fφðyÞ: y < x, y ∈ Cg for x ∉ C. Then, φ is continuous and monotonically increasing. Hence, there exists a positive Borel measure μ on T such that The measure μ (the so-called Cantor middle-third measure) is a typical example of a singular continuous measure. We refer the reader to the papers [24] and [25] which treat measures of the Cantor type. It is known that Hence, Since 0 ≤ sin 2 ð2πn/3 j Þ < 1 for each j and ∑ ∞ j=1 sin 2 ð2πn/ 3 j Þ < ∞, it follows that b μðnÞ ≠ 0. Note also that b μð−nÞ = b μðnÞ and b μð3nÞ = b μðnÞ for every n ∈ ℤ. We may here ask the following questions: (a) What is DðT μ Þ? Is DðT μ Þ dense in H 2 ðDÞ?
(b) What is T μ ? Is T μ trivial?
We next ask: when is the domain DðT μ Þ dense in H 2 ðDÞ ? It does not seem easy to answer this question in general. The following lemma is used to derive some properties of DðT μ Þ which are helpful to determine the density of DðT μ Þ in H 2 ðDÞ. Recall that S is the shift operator on HðDÞ, i.e., if f ∈ HðDÞ, then Sf ðzÞ = zf ðzÞ for z ∈ D.
Conversely, suppose that f ∈ H 2 ðDÞ and ðS − αÞf ∈ cl H 2 ð DðT μ ÞÞ. Then, there exists a sequence fg j g in DðT μ Þ such that We want to show that f ∈ cl H 2 ðDðT μ ÞÞ. To see this we consider two cases. Case 1. (|α | <1). Assume first that g j ðαÞ = 0 for all j. Then, where f j ∈ C A ðDÞ. Since g j ∈ DðT μ Þ, it follows from (a) that f j ∈ DðT μ Þ. Note that the approximate point spectrum of the operator S on H 2 ðDÞ is σ ap ðSÞ = T (cf. [26]). Since α does not belong to T , the operator S − α is bounded below on H 2 ðDÞ. It follows that there exists a constant c > 0 such that for all j. This implies that ∥f − f j ∥ 2 → 0. Therefore, f ∈ cl H 2 ðDðT μ ÞÞ.

Journal of Function Spaces
In the case that g j ðαÞ ≠ 0 for some j, we may assume that g 1 ðαÞ ≠ 0. Note that g j ⟶ ðS − αÞf weakly. Hence, g j ðzÞ ⟶ ððS − αÞf ÞðzÞ for each z ∈ D. In particular, we have Now put Then, h j ∈ DðT μ Þ and h j ðαÞ = 0 for all j. Observe that It follows that Hence, by the preceding paragraph, we conclude that f ∈ cl H 2 ðDðT μ ÞÞ.
As a consequence of Proposition 6, we derive the following theorem which describes the domain DðT μ Þ. Recall that an inner function is said to be singular if it has no zero in the unit disk. Theorem 8. Let μ ∈ MðT Þ. Then, one of the following holds: where θ is an inner function or θ = 0. If θ = 0, then the case (i) occures. If θ is a nonzero constant function, case (ii) occurs. Now, suppose that θ is nonconstant. We show that θ has no zero in D. To see this, choose any nonzero function f in DðT μ Þ. Fix an arbitrary point α of D and let n be the multiplicity of the zero of f at α. Then, where g ∈ C A ðDÞ and gðαÞ ≠ 0. Hence, by a repeated application of Proposition 6(a), we have It follows that g = θh for some h ∈ H 2 ðDÞ. Thus, θðαÞ cannot be 0. Since α was arbitrary, we conclude that θ has no zero in D. Therefore θ is a singular inner function.
Remark 9. Unfortunately, we cannot find a concrete example for the third case. It would be possible that the third case never occurs.
The following proposition is another consequence of Proposition 6 which gives a sufficient condition for the domain DðT μ Þ to be dense in H 2 ðDÞ.
Proof. Suppose that cl H 2 ðDðT μ ÞÞ contains a polynomial. Then, by Proposition 6, (b), there exists a polynomial p ∈ c l H 2 ðDðT μ ÞÞ, all of whose zeros are in T , such that pð0Þ = 1. Let ζ 1 , ⋯, ζ N ∈ T be the zeros of p, listed according to their multiplicities. Then, Choose a sequence fk n g in ℕ such that k n+1 > Nk n (e.g., k n = ðN + 1Þ n ). For each n ∈ ℕ, define All of them are polynomials, divisible by p. Since cl H 2 ðDðT μ ÞÞ is S-invariant, the polynomials p n belong to DðT μ Þ. It follows by a direct computation that for every n ∈ ℕ. This implies that p n ⟶ 1 in H 2 ðDÞ.

6
Journal of Function Spaces Therefore, the constant function 1 belongs to cl H 2 ðDðT μ ÞÞ.
Remark 11. Proposition 10 shows that the domains DðT μ Þ, presented in (a) and (b) of Example 4, are dense in H 2 ðDÞ, because they contain the polynomial pðzÞ = 1 − z. The proof of Proposition 10 shows that every polynomial, all of whose zeros are in T , is an outer function.
In order to consider the matrix representation of a linear operator on H 2 ðDÞ, it is necessary that its domain contains all polynomials. Let us interpret the condition that DðT μ Þ contains all polynomials. Note that this is equivalent to the condition that DðT μ Þ contains any polynomial which does not vanish on T , by Proposition 6, (a).
The converse is a part of Proposition 2.
On the other hand, we would like to conjecture the following:

Conjecture 14. Every Toeplitz operator with a singular symbol is trivial.
We give evidence for Conjecture 14 by using the known fact about the Cauchy transform. Let E be a closed subset of T and let Then, it is known that FðEÞ = f0g if and only if mðEÞ = 0 (cf. [23], Theorem 5.5.2).
The Cantor-middle-third measure μ in Example 4, (c), is a singular continuous measure, and its support is the Cantor set (in T ) whose Lebesgue measure is 0. Hence, Theorem 15 implies that T μ is trivial.
We have seen that the Toeplitz operator T μ in Example 4, (b), is a densely defined trivial linear operator. This result can be extended to the case that μ has a finite support. In this case, the fact that T μ is trivial may follow from Theorem 15. However, we give a direct proof and also show that T μ is densely defined.

Proposition 17.
Let μ ∈ MðT Þ be a discrete measure whose support is a finite set. Then, the Toeplitz operator T μ is a densely defined trivial linear operator with domain Proof. Suppose that sup pμ consists of N distinct points ζ 1 , ⋯, ζ N of T . Then, where c 1 , ⋯, c N are nonzero complex numbers and δ ζ is the unit point mass concentrated at ζ. We first show that For any f ∈ C A ðDÞ, It follows that Conversely, let f ∈ DðT μ Þ. Then, Pðf · μÞ ∈ H 2 ðDÞ. For each j, put Then, F = ∑ N j=1 F j is the nontangential limit function of Pð f · μÞ. Thus, F ∈ H 2 ðT Þ. Choose disjoint open arcs I j ⊆ T with ζ j ∈ I j . Fix an index j 0 and let χ denote the characteristic function of I j 0 . Then, χ · F ∈ L 2 ðT Þ. Also, χ · F j ∈ L ∞ ðT Þ for each j ≠ j 0 . Hence, Since ð1 − χÞ · F j 0 ∈ L ∞ ðT Þ, it follows that This implies that f ðζ j 0 Þ = 0, because otherwise, F j 0 ∉ L 2 ð T Þ. Since j 0 was arbitrary, we have f ðζ j Þ = 0 for each j. It follows that This proves (62). In particular, DðT μ Þ contains the polynomial pðzÞ = ðz − ζ 1 Þ ⋯ ðz − ζ N Þ. Hence, by Proposition 10, DðT μ Þ is dense in H 2 ðDÞ.
Example 18. Let μ ∈ MðT Þ be a discrete measure whose support has only finitely many limit points, for example, where ζ n = e πi/2 n . By an argument similar to the proof of Proposition 17, we may show that D T μ À Á = f ∈ C A D ð Þ: f ζ ð Þ = 0 for every ζ ∈ sup pμ f g , ð70Þ and T μ f = 0 for all f ∈ DðT μ Þ. Hence, T μ is trivial. Note that every polynomial has only finitely many zeros. It follows that DðT μ Þ cannot contain any polynomial. Nevertheless, DðT μ Þ contains a nonzero function by Fatou's theorem for C A ðDÞ, which says that, for any given closed set K ⊆ T with mð KÞ = 0, there exists a function in C A ðDÞ which vanishes precisely on K (cf. [19]). Hence by Theorem 8, DðT μ Þ is dense in H 2 ðDÞ or cl H 2 ðDðT μ ÞÞ = θH 2 ðDÞ for some singular inner function θ. But it does not seem easy to determine whether DðT μ Þ is dense in H 2 ðDÞ or not.
To each Toeplitz operator T μ , there corresponds an infinite Toeplitz matrix Tðb μÞ. In general, however, it is a bit awkward to call Tðb μÞ as the matrix representation of T μ , because the domain DðT μ Þ may not contain the monomials z n . Nevertheless, often, information about T μ gives information about Tðb μÞ. The following is one of such example. 8 Journal of Function Spaces Corollary 19. Let μ ∈ MðT Þ be a discrete measure whose support consists of N points of T . Then, for all n ≥ N.