Skew-symmetric and essentially unitary operators via Berezin symbols

Abstract We characterize skew-symmetric operators on a reproducing kernel Hilbert space in terms of their Berezin symbols. The solution of some operator equations with skew-symmetric operators is studied in terms of Berezin symbols. We also studied essentially unitary operators via Berezin symbols.


Introduction
We denote by a complex separable infinite dimensional Hilbert space endowed with the inner product 〈⋅ ⋅〉 , , and ( ) the algebra of all bounded linear operators on . Sometimes the letter will denote the so-called reproducing kernel Hilbert space (RKHS) over some set Ω. Also, for any self-adjoint operator T on the operator iT is skew-symmetric.
As it is known, many classical results in the matrix theory deal with complex symmetric matrices (i.e., = T T t ) and skew-symmetric matrices (i.e., = − T T t ). These concepts appear naturally in a variety of applications such as complex analysis, functional analysis (including operator theory), and even quantum mechanics.
In [1], Zagorodnyuk studied the polar decomposition of skew-symmetric operators and obtained some basic properties of skew-symmetric operators. Later, Li and Zhu [2] also noted that an important way to investigate the structure of skew-symmetric operators is to characterize the skew-symmetry of concrete class of operators. For example, Zagorodnyuk [3] studied the skew-symmetry of cyclic operators. Li and Zhu [4] studied the skew-symmetry of normal operators and gave two structure theorems of skew-symmetric normal operators. For more information about skew-symmetric operators, see, for instance, Zhu [5], Li and Zhu [6], and references therein.
In the present article, we characterize skew-symmetric operators on an RKHS in terms of their Berezin symbols. We also study in terms of the Berezin symbols the solvability of some operator equations with skew-symmetric operators. Note that such an approach, apparently, was initiated by the second author in [7]. We also give in terms of the Berezin number a sufficient condition providing essential unitarity of essentially invertible operators.
2 Reproducing kernel Hilbert space and some properties A reproducing kernel Hilbert space is a Hilbert space = ( ) Ω of complex-valued functions on some set Ω such that evaluation → ( ) f f λ at any point of Ω is a continuous functional on . The Riesz representation theorem ensures that an RKHS has a reproducing kernel, that is, for every ∈ λ Ω there is a unique element for all ∈ f . We call the function k λ , the reproducing kernel at λ. The following proposition gives a way to compute the reproducing kernels (see, for instance, Aronzajn [8], Halmos [9], and Stroethoff [10] where the convergence is in . In particular, It follows from the aforementioned proposition that The function is called the normalized reproducing kernel at λ.
Definition 2.1. Let T be a bounded linear operator on , the Berezin symbol of T is defined by Note that the Berezin symbol  T is a complex-valued bounded function, because  denote the Berezin number of T defined by Thus, T is uniquely determined by the function This is the case, for example, whenever the functions ( ) T z λ , are holomorphic in z and λ , by a wellknown classical theorem in complex analysis (see, for example, Folland [11] and Stroethoff [12,Lemma 2.3]). The following uniqueness lemma is due to the second author (see [13,Lemma 2]).

Lemma 2. Let
= ( ) Ω be an RKHS whose elements are functions on some set Ω. If satisfies condition (*), then the correspondence  ↔ T T is one-to-one, i.e., = T 0 if and only if  ( ) = T λ 0 for all ∈ λ Ω.
Definition 2.2. The RKHS = ( ) Ω is said to be standard (see Nordgren and Rosenthal [14]) if the underlying set Ω is a subset of a topological space and the boundary ∂Ω is nonempty and has the property that , n converges weakly to 0 whenever { } λ n is a sequence in Ω that converges to a point in ∂Ω.
The common Hardy, Bergman, and Fock Hilbert spaces are standard in this sense (see Stroethoff [10]).
For a compact operator K on the standard RKHS , it is clear that  ( ) = →∞ K λ lim 0 n n whenever { } λ n converges to a point in ∂Ω, since compact operators send weakly convergent sequences into strongly convergent ones. In this sense, the Berezin symbol of a compact operator on a standard RKHS vanishes on the boundary.

Operator equations on * C -algebras
The equations = AX C and = XB D for operators, including square and rectangular matrices, have a long history. In particular, the first equation has applications in the control theory. For more information and application about these equations, see, for example, [15] by Dajic and Koliha. In the present section, we study the solution of the operator equations = + TX K Y and = + XT L (where K, L are compact and Y, are skew-symmetric operators) in some Engliš * C -operator algebras of operators on the reproducing kernel Hilbert spaces, including the Hardy space = ( ) H H 2 2 over the unit disc of the complex plane . First, we need some notations and preliminaries.
The Hardy space is the Hilbert space consisting of the analytic functions on the unit disc Since { } ≥ z n n 0 is an orthonormal basis in H , 2 it is easy to see from Proposition 1 that by associating to each function ∈ f H 2 its radial boundary values ( )( ) ≔ ( ) which (by the Fatou Theorem [16]) exist for almost all ∈ = ∂ ζ , where m is the normalized Lebesgue measure on . Then we have is the Riesz projection (orthogonal projection). The harmonic extension of function ∈ ∞ φ L is defined by  φ : The following lemma is well known (see, for instance, Engliš [17] and Zhu [19]).
The main lemma for our further discussions is the following lemma due to Engliš [ of H 2 are, loosely speaking, asymptotic eigenfunctions for any Toeplitz operator for almost all ∈ ) t π 0, 2 .
The following set is defined by Engliš in [17, (6) in Section 3] We call ε H 2 the Engliš algebra of operators on the Hardy space H . 2 It follows from Lemma 4 that for any φ H 2 Here we study the solution of the operator equations = + and in the Engliš algebra ε , where K is compact and Y is a skew-symmetric operator, which is defined on an RKHS = ( ) Ω , as follows: As in the case = H , 2 it can be shown that actually the set ε is a * Calgebra (see the proof of (A1) in Section 3 of Engliš's paper [17]). We also characterize the skew-symmetric operators in terms of Berezin symbols.
and hence for all ∈ λ Ω. Then we have from which we have that for all ∈ λ Ω. On the other hand, by considering that and hence for all ∈ λ Ω. This shows that So, since we have from the inequality (4) that and therefore, for all ∈ λ Ω. Now the result follows immediately from this inequality. This proves (ii). (iii) It follows from (3) that for all ∈ λ Ω, and since ⊥ * X T , we have for all ∈ λ Ω. This implies by Lemma 2 that as desired. This proves the theorem. □

Remark 1. Assertion (ii) in Theorem 3 shows that the necessary condition
is not in general a sufficient condition for a solution X of (2) being from the class .
Corollary 7. Let ( ) Ω be an RKHS such that the functions ( ) T z λ , satisfy condition (*) whenever T is a bounded linear operator on . Let ∈ ( ) T K Y , , be operators such that Y is skew-symmetric and K is not skew-symmetric. Then the operator equation = + TX K Y has no solution with the property that ( )⊥ ( ) * Range X Range T .

Remark 2.
It is easy to see from the proof of Theorem 6 that the same results can be obtained for the operator equation = + XT K Y; we omit them.

Proposition 8. Let = ( )
Ω be an RKHS, and let ∈ ( ) T T K K Y Y , , , , , 1 and Y 2 are the skew-symmetric operators. If ∈ X ε satisfies the equations = + T X K Y 1 1 1 Proof. The proof is similar to the proof of (i) in Theorem 6. Indeed, if ∈ X ε is a common solution of equations (5) and (6), then as in the proof of (i) in Theorem 6, we obtain that Since ∈ X ε and  ( )  (5) and (6), then The proof of the following corollary is immediate from Lemmas 3 and 4 and Proposition 8.
is a common solution of (5) and (6), then  a λ  a λ  a λ  a λ  a λ  ,  ,  , , , , , we obtain the formula where ( ) = A z d v H . This implies that equals the closed subalgebra of ( ) L a 2 generated by the Toeplitz operators with bounded harmonic symbol, and that also equals the closed subalgebra of ( ) L a 2 generated by { ∈ } T u : u . The main goal of the paper [20] is to study the boundary behavior of the Berezin symbols of the operators in and of the functions in . Namely, the author's study shows (see [20,Theorem 2.11]) that if ∈ S , then ∈ ∼ S . Also, they prove (see [20,Corollary 3.4]) that if ∈ u , then − ∼ u u has nontangential limit 0 at almost every point of ∂ . Using similar techniques, they prove (see [20,  , 2 has nontangential limit 0 at almost every point of ∂ . Since this property is mainly used in the proof of Corollary 10, the same results also can be proved for the Bergman space Toeplitz operators T u with ∈ u , which we omit.

Remark 4.
As an application of Proposition 5 and also for its usefulness note the following: if ∈ ∞ φ H is a non-constant function and θ is a nontrivial analytic map of onto itself (i.e., ( ) ⊂ θ and ≠ θ constant), then we can construct a weighted composition operator with symbols φ θ , , for f in the Hardy space H 2 , where T φ is an analytic Toeplitz operator and C θ is a composition operator on H 2 .
Note that = * T T φ φ , co-analytic Toeplitz operator. However, in the theory of composition operator determination of the adjoint is a problem of some interest. For example, this question is not trivial even for the composition operator C z 2 ; for more discussion about the adjoint of composition operators, see [21][22][23][24][25][26] and references therein. Thus, in particular, the investigation of skew-symmetric weighted composition operators and the operators of the form * C T θ φ with ∈ ∞ φ L is not in general a trivial question, while for the symbol θ with ( ) = θ 0 0, C θ is obviously non skew-symmetric. In fact, since the set { ∈ } k λ : But, since ( ) = θ 0 0, for = λ 0 this equality does not hold, and hence ≠ − * C C θ θ , that is, C θ is not skew-symmetric.
More generally, every composition operator C θ on H 2 (or on the Bergman space ( ) L a 2 ) is not skewsymmetric. In fact, for = λ 0 we have that ), which is impossible. This shows that every composition operator C θ on the Hardy and Bergman spaces is not skew-symmetric. However, in the following examples we demonstrate usefulness and application of Proposition 5, since * C k θ λ has an explicit expression: Indeed, for any ∈ μ , we have: Proof. Let ∈ λ be arbitrary. Then by using formula (7), we have:

on .
Proof. Indeed, we have:  is the orthoprojection. When = ρ 1 and = ρ 2, this definition reduces to the operator norm and numerical radius, respectively. (For more details see also Garayev [7], and Sahoo et al. [30]. ) We say that an operator ∈ ( ) T is essentially invertible (or Fredholm) if there exists an operator ∈ ( ) A such that − AT I and − TA I are both compact operators, i.e., = + = + AT I K TA I K and 1 2 for some compact operators K 1 , K 2 on H. An essential inverse of T will be denoted as − T ess 1 . Thus, the following problem naturally arises.

Problem 1
To find in terms of Berezin numbers of an essentially invertible operator T and its essential inverse − T ess 1 the necessary and sufficient conditions under which T be an essentially unitary operator on RKHS = ( ) Ω (i.e., = + * T T I K 1 and = + * TT I K 2 for some compact operators K 1 and K 2 ). Note that the class of essentially unitary operators has not been extensively studied with respect to the class of unitary operators.
The present section, which is motivated by this question (see also [7]), gives in terms of the Berezin numbers of operators * TT and ( ) * − TT ess 1 a sufficient condition for essentially unitarity of the essentially invertible operator T on the RKHS. Our result improves the result of the paper [7, Theorem 1] where only the case = = K K 0 1 2 is considered.
Theorem 11. Let = ( ) Ω be a standard RKHS with the property (*) (Section 2) and ∈ ( ) T be an essentially invertible operator, associated with the compact operators K 1 and K 2 . If