Phillips symmetric operators and functional calculus of maximal symmetric operators

The aim of this work is to develop a functional calculus for simple maximal symmetric operators. The proposed approach is based on the properties of self-adjoint extensions of Phillips symmetric operators. The obtained results are applied to the description of non-cyclic vectors of backward shift operators.


Introduction
Let S be a symmetric operator acting in a Hilbert space H and let U be a family of unitary operators in H such that the inclusion U ∈ U implies U * ∈ U. The operator S is called U-invariant if S commutes with all U ∈ U. Does there exist at least one Uinvariant self-adjoint extension of S? The answer is affirmative if S is a semibounded operator and the Friedrichs extension of S gives the required example.
In the general case of non-semibounded operators, Phillips constructed a symmetric operator S and a family U of unitary operators commuting with S such that S has no U-invariant self-adjoint extensions [14, p. 382]. It was discovered by Kochubei [7] Communicated by Ilya Spitkovsky. that the characteristic function of the symmetric operator 1 constructed in the Phillips work is a constant in the upper half-plane C + . This observation was used in [9] for the general definition of Phillips symmetric operators. Namely, we say that a closed densely defined symmetric operator with equal defect numbers is a Phillips symmetric operator (PSO) if its characteristic function is an operator-constant on C + .
Self-adjoint and, more generally, proper extensions of PSO possess a lot of curious properties [3,7,9,10]. Part of them is used for the development of the functional calculus of simple maximal symmetric operators 2 in the present paper.
The functional calculus for simple maximal symmetric operators was announced by Plesner in two short papers [15,16] without proofs. To the best of our knowledge, these papers have not been translated.
In the present paper, we propose an approach to the functional calculus that is based on the properties of self-adjoint extensions of PSO (Theorems 2.2, 2.4) and it gives rise to the all-around development of Plesner's ideas. Such kind of functional calculus turns out to be useful in the Lax-Phillips scattering theory [5,6].
Let B be a simple maximal symmetric operator in H. In Sect. 3 we define an operator ψ(B) for each Lebesgue measurable function ψ and investigate its properties (Proposition 3.2, Theorem 3.3). The results are simplified if ψ ∈ H ∞ . In this case, Plesner's functional calculus is reduced to functional calculus for the special class of self-adjoint operators (Theorem 3.4, Corollaries 3.5, 3.6). Further, we discuss a relationship of our results with functional calculus for unilateral shifts which is a special case of functional calculus for completely nonunitary contractions [18,Chapter III].
Special attention is paid to the case of inner functions ψ (Proposition 3.7, Corollary 3.8). An application of the functional calculus to the description of non-cyclic vectors of the backward shift operator is considered (Propositions 3.10, 3.11).
Throughout the paper D(A), R(A), ker A, ρ(A), and σ (A) denote the domain, the range, the null-space, the resolvent set, and the spectrum of a linear operator A, respectively, while A| D stands for the restriction of A to the set D. The continuous spectrum σ c (A) of a linear operator A consists of λ ∈ σ (A) for which there exists a non-compact sequence { f n } such that f n = 1 and lim n→∞ (A − λI ) f n = 0.
A subspace K of a Hilbert space H is said to be an invariant subspace of the operator where P K is a orthogonal projection onto K and the subspaces K and K ⊥ = H K are invariant for A.
The symbols H p (D) and H p (C + ) are used for the Hardy spaces in D = {λ ∈ C : |λ| < 1} and C + = {z ∈ C : Im z > 0}, respectively. The Sobolev space is denoted as W p 2 (I ) (I ∈ {R, R + = [0, ∞)}, p ∈ {1, 2}). The notations H p (D, N ) and H p (C + , N ), and W p 2 (I , N ) are used for the Hardy and Sobolev spaces of vector functions with values in an auxiliary Hilbert space N . The symbol < n > means linear span of an element n ∈ N , while α X α means the closure of the linear span of the sets (vectors) X α .

Simple maximal symmetric operator
Let B be a densely defined symmetric operator in a Hilbert space H with inner product (·, ·) linear in the first argument. The defect numbers of B in C + and C − are defined as A closed symmetric operator B is called maximal symmetric if one of its defect numbers is equal to zero. If B is a maximal symmetric operator, then A closed symmetric operator B is simple if there is no subspace that reduces it and on which it induces a self-adjoint operator.
One of the simplest examples of a simple maximal symmetric operator is the first derivative operator considered in L 2 (R + , N ) where

Phillips symmetric operators
The original definition of PSO deals with the concept of the characteristic function. For our considerations, it is convenient to use an equivalent definition established in [9]. It follows from the definition above and the relations (2.3), (2.4) that every simple PSO S is unitarily equivalent to the symmetric momentum operator with one-point interaction The following two theorems are principal for our presentation.
Theorem 2.2 [3,9] Self-adjoint extensions of a PSO S are unitarily equivalent to each other. If S is a simple PSO, then its self-adjoint extensions are unitarily equivalent to the self-adjoint momentum operator in L 2 (R, N ) That gives B u ∈ H . Therefore, B is a symmetric operator in H .
In view of (2.9), Consider a simple symmetric operator The self-adjoint momentum operator A acting in L 2 (R, N ) and defined by (2.6) is an example of a minimal self-adjoint extension of a simple maximal symmetric operator B defined by (2.2). Applying the Fourier transformation 3 we obtain FA = MF i.e., A is unitarily equivalent to the multiplication operator M defined by (2.7).

Definition of Ã(B)
Let B be a symmetric operator in a Hilbert space H. A family of bounded operators According to Naimark's results [2, Section 9], each symmetric operator possesses at least one spectral function and every spectral function {E δ } of a symmetric operator B has the form It is important that a spectral function of a maximal symmetric operator is determined uniquely 4 [2, p. 402]). For this reason, similarly to the self-adjoint case, one can try to define operators for some class of functions ψ. To do that we consider an operator of multiplication by a Lebesgue measurable function ψ in the space L 2 (R, N ). It can be presented as a function ψ(M) of the multiplication operator M by an independent variable in where the operators E M δ of the spectral function of M act as the multiplication by characteristic function χ (−∞,δ] of the intervals (−∞, δ].
Let an operator A in H be a minimal self-adjoint extension of a simple maximal symmetric operator B acting in H ⊂ H. In view of Theorem 2.4, A is unitarily equivalent to M, i.e., there exists a unitary mapping G : This means that the operator is well defined and By virtue of (3.1) and (3.2), where f ∈ D(ψ(A)) ∩ H and g ∈ H. Therefore, for a simple maximal symmetric operator B, the operator ψ(B) in (3.2) can be defined as follows  Since the inclusion (3.9) holds if 1 δ−μ n ∈ D(ψ(M)). The last relation is equivalent to (3.12) It is easy to see that (3.12) holds when ψ ∈ L p (R) and p ≥ 2.

Spectral properties of Ã(B)
The formula (3.5) defines ψ(B) as the projection of ψ(A). The functional calculus of self-adjoint operators states that the spectrum of ψ(A) is determined by the range of the function ψ considered on the spectrum of A. In our case, the operator A is a selfadjoint extension of a simple PSO (2.10). For this reason, its spectrum coincides with R (it follows from Theorem 2.4). Hence, one may expect that the spectrum of ψ(B) will be determined by the range of ψ. To prove the corresponding result (Theorem 3.3) we start with the auxiliary notations and results from [13].
The operator where P + is an orthogonal projection operator in L 2 (T) onto the subspace H 2 (D) is called a Toeplitz operator with the symbol φ. The operator T φ is bounded in H 2 (D). The essential range of a symbol φ is defined as follows: A Toeplitz operator in H 2 (C + ) with the symbol ψ ∈ L ∞ (R) has the form where P + is an orthogonal projection operator in L 2 (R) onto the subspace H 2 (C + ). The essential range of ψ is: If the symbols φ ∈ L ∞ (T) and ψ ∈ L ∞ (R) of the Toeplitz operators T φ and T ψ satisfy the relation then T φ and T ψ are unitarily equivalent [13, p. 261].
and relations (2.3) and (3.7) we arrive at the conclusion that G = F maps unitarily the subspace H of H onto the subspace H 2 (C + , N ) of L 2 (R, N ) and the orthogonal projection operator P in H onto H has the form P = G −1 P + G, where P + is an orthogonal projection in L 2 (R, N ) onto H 2 (C + , N ). Summing up, the formula (3.5) can be rewritten as follows: where H 2 (C + , n ) denotes a Hardy subspace of functions with values from n . By the construction, each subspace H 2 (C + , n j ) reduces the operator P + ψ(M) and its restriction onto H 2 (C + , n j ) is unitarily equivalent to the Toeplitz operator T ψ in H 2 (C + ). Therefore, the operator P + ψ(M) in H 2 (C + , N ) is unitarily equivalent to the orthogonal sum of operators T ψ ⊕ T ψ ⊕ · · · ⊕ T ψ dim N terms acting in the space H 2 (C + ) ⊕ H 2 (C + ) ⊕ · · · ⊕ H 2 (C + ). As a consequence, ψ(B) is unitarily equivalent to the orthogonal sum of Toeplitz operators T φ acting in H 2 (D) where functions ψ and φ satisfy (3.13). Now, taking into account the above consideration, we can justify statements of the theorem using the corresponding results for T φ given in [13]. Precisely, in view of [13,

Functional calculus
Actually, (3.5) allows one to define ψ(B) for an arbitrary Lebesgue measurable function ψ. This formula can be essentially simplified if ψ ∈ H ∞ . The next result was proved in [5]. For the reader's convenience, the principal stages of the proof are presented. N ) is an invariant subspace for the operator of multiplication ψ(M) acting in L 2 (R, N ). Recalling that G = F maps H onto H 2 (C + , N ) (see the proof of Theorem 3.3) one conclude that H is an invariant subspace for the operator G −1 ψ(M)G in H. Now, one can rewrite (3.15) as follows:

Theorem 3.4 Let B be a simple maximal symmetric operator with m + (B) = 0 acting in a Hilbert space H and let A be a minimal self-adjoint extension of B in H ⊃ H. Then
that completes the proof. Corollary 3.5 Let B be a simple maximal symmetric operator with m + (B) = 0 and let ψ ∈ H ∞ (C + ). Then the following statements are true: = 1 a.e., then ψ(B) is an isometric operator in H; Proof Follows from Theorem 3.4 and functional calculus of self-adjoint operators. Then, for all f ∈ ker(B * − μI ) where μ ∈ C − , the following relation is true (3.16) where B is defined by (2.2) and ψ(B) = F −1 ψ(M)F. Therefore, ψ(B) and ψ(B) are unitarily equivalent and, without loss of generality, one can consider the case, where H = L 2 (R + , N ) and B = B.

Proof It follows from (3.7), (3.8) and Theorem 3.4 that
By virtue of Corollary 3.5, for arbitrary f ∈ ker(B * − μI ) and u ∈ D(B), This means that ψ(B) * f is orthogonal to R(B − μI ) and hence, ψ(B) * f belongs to ker(B * − μI ). Recalling now (3.10), we arrive at the conclusion that where n ∈ N is determined uniquely by the vector f = e −iμx n. Relation (3.17) and the fact that the subspace L 2 (R + , n ) is a reducing subspace for the operator ψ(B) = F −1 ψ(M)F acting in L 2 (R + , N ) lead to the conclusion that Hence, the vector n in (3.17) has the form n = cn, where a constant c ∈ C should be specified. To do that, one first calculates (3.18) where (·, ·) N means a scalar product in N .
On the other hand, where (see (3.11)) After the comparison of relations (3.18) and (3.19) we get The application of the Poisson formula [13, p. 147] leads to the equality that completes the proof. (3.21)

Relationship with unilateral shifts
The above-mentioned one-to-one correspondence between B and T can be easily deduced from [18,Chapter III,§9]. A unilateral shift is an example of a completely nonunitary contraction. For such operators, the functional calculus is well-developed [18]. This gives rise to an idea to define the operator ψ(B) with the use of the Cayley transform T of B.
Let A be a minimal self-adjoint extension of B acting in H. Its Cayley transform is a unitary operator in H and, according to [18, p. 117], where P is the orthogonal projection operator in H on H.
where ψ(δ) = φ( δ−i δ+i ) belongs to H ∞ (C + ). Using (3.5) and (3.22) we arrive at the conclusion that The obtained relationship allows one to reduce the investigation of ψ(B) to the investigation of φ(T ). Proof In view of (3.23), it is sufficient to prove that φ(T ) is a unilateral shift in H with the wandering subspace L = ker φ(T ) * .

The case of inner functions
Due to the relation ψ(δ) = φ( δ−i δ+i ), the function φ belongs to H ∞ (D) and it is a non-constant inner function. Therefore, |φ(λ)| < 1 for λ ∈ D [12, p. 49]. Taking into account that the unitary shift T is an example of completely nonunitary contraction and using [18, Chapter III, Theorem 2.1 (e)] we arrive at the conclusion that φ(T ) is also a completely nonunitary contraction in H. Simultaneously, φ(T ) is an isometric operator (it follows from (3.23) and Corollary 3.5). In view of the Wold decomposition Assume now that f ∈ D(Bψ(B)) and consider the operator Elementary analysis with the use of Corollary 3.5 and the relation Y * Y = I gives rise to the following conclusions: It follows from (a)-(e) that C is a symmetric extension of the maximal symmetric operator B in H. This means that f ∈ D(B) and the inverse inclusion Bψ(B) ⊆ ψ(B)B is proved. In this case, one should consider functions ψ from H ∞ (C − ) and choose μ ∈ C + in Corollary 3.6.

Non-cyclic vectors
Let T be a unilateral shift in H. It's adjoint operator T * is called a backward shift operator. A vector f ∈ H is called non-cyclic for the backward shift operator T * if where ψ ∈ H ∞ (C + ) runs the set of inner functions and a simple maximal symmetric operator B is defined by (3.21).
To prove an inverse inclusion we note that the operator ψ(B) is unitarily equivalent to ψ(B) (see (3.16)) and, without loss of generality, one can consider the case, where: Since the multiplicity of T is 1, the space N can be chosen as C. In this case, L 2 (R + , C) = L 2 (R + ) and for f ∈ M T * , the subspace T * n f turns out to be an invariant subspace for T in L 2 (R + ). Applying the Fourier transformation (2.11) to the relation above and using (3.14) we conclude that the subspace F[L 2 (R + ) E f ] of H 2 (C + ) is invariant for the operator of multiplication by δ−i δ+i in H 2 (C + ). By virtue of Beurling's theorem [12, p. 49] there exists an inner function ψ ∈ H ∞ (C + ) such that F[L 2 (R + ) E f ] = ψ(δ) H 2 (C + ). Therefore, where M is defined by (2.7). The obtained relation means that f is orthogonal to ψ(B)L 2 (R + ). Therefore, f ∈ ker ψ(B) * and ker ψ(B) * ⊇ M T * . The relation (3.26) is proved.