Weighted Shifts on Directed Semi-Trees: an Application to Creation Operators on Segal–Bargmann Spaces

We extend the notion of a weighted shift on a directed tree to the case of a more general graph which we call a directed semi-tree. Some basic properties of such operators are investigated. It is shown that a generalized creation operator on the Segal–Bargman space is unitarily isomorphic to a weighted shift on a directed semi-tree of a particular form.


Introduction
Relating Hilbert space operators to directed graphs can provide some efficient new tools in operator theory. The study of non-selfadjoint operators can be, for example, successfully realized through connecting directed graphs with corresponding adjacency bounded operators. Such an approach appeared in [8] and was subsequently developed in [7]. In order to obtain significantly more information about operators  related to graphs Jabłoński, Stochel, and Jung focused attention in [9] on directed trees. They introduced the notion of a weighted shift on a directed tree and effectively investigated properties of a class of such (unbounded) operators. We refer to [4][5][6][10][11][12] for further results on the subject. It should be noticed that this concept substantially generalized the notion of a classical weighted shift (see, for example, [15] or [13]). In the present paper we extend the definition of a weighted shift on a directed tree from [9] to the case of a more general graph which we propose to call a directed semi-tree. The idea of broadening this definition to a larger class of operators was motivated by the fact that the properties of generalized creation operators on Segal-Bargman spaces cannot be described by using the theory developed in [9]. Here we obtain counterparts of some results included therein in this new setting. It is worth noting that most of arguments of [9,Chapter 3] do not transfer automatically to the case of directed semi-trees. Therefore, in the paper we have to apply some alternative proof techniques. It turns out that our results embrace generalized creation operators.
The paper is organized as follows. In Sect. 2 we set up notation and terminology of the graph theory which is used throughout this work. We recall some properties of directed trees and next introduce a concept of a directed semi-tree. In Sect. 3 we define n-finite and infinite Bargmann graphs, discuss their properties, and show that they are in fact directed semi-trees. In Sect. 4 we show that Segal-Bargmann spaces can be regarded as l 2 -Hilbert spaces defined on vertices of the Bargmann graphs. The main results of the paper are to be found in Sects. 5-7. In Sect. 5 we generalize the notion of a weighted shift on a directed tree from [9] to the case of a directed semi-tree, and next we prove that such weighted shifts have, in some situations, similar properties to those on trees. Section 6 is devoted to showing that the generalized creation operators, defined by Bargmann in [2] and [3], are unitarily isomorphic to weighted shifts on the respective Bargmann graphs. Finally, in Sect. 7 we give some sufficient conditions for the orthonormal basis of the l 2 -type space to be a core for a weighted shift on a directed semi-tree. We also observe that these conditions are fulfilled by the generalized creation operators. The differences between weighted shifts defined on directed trees and those defined on directed semi-trees are highlighted in Sects. 5 and 7. The inspiration for writing this paper comes from [9] and [17].

Directed Trees and Directed Semi-Trees
Throughout the paper, we employ the standard terminology of the graph theory which was also utilized in [9].
We say that a pair G = (V, E) is a directed graph if V is a nonempty set and E is a subset of (V × V )\{ (v, v) : v ∈ V }. Put We say that a member of V is a vertex of G, a member of E is an edge of G, and finally a member of E is an undirected edge of G. If W is a nonempty subset of V , then the pair (W, E ∩ (W × W )) is also a directed graph which is called a directed subgraph of G. A directed graph G is said to be connected if for every two distinct vertices u and v of G there exists a finite sequence v 1 , . . . , v n (n ≥ 2) of vertices of G such that u = v 1 , {v j , v j+1 } ∈ E for all j = 1, . . . , n − 1, and v n = v. Such a sequence is called an undirected path joining u and v. Let We say that a member of Chi(u) is a child of u and a member of Par(v) is a parent of v. If for a vertex v ∈ V there exists a unique vertex u ∈ V being its parent (i.e. ) is an edge of G. We denote by Root(G) the set of all roots of G. If Root(G) is a one-element set, then its unique element is denoted by root. For W ⊂ V , we set W • = W \Root(G) and write card(W ) for the cardinality of W .

Proposition 2.1
Let G be a directed graph satisfying the following conditions: Then the set Root(G) contains at most one element.
Let G be a directed graph. Given a set W ⊂ V , we put Define also Chi 0 (W ) = W, Chi n+1 (W ) = Chi(Chi n (W )) for n = 0, 1, . . . , and Chi n (W ). Let us recall some properties of directed trees given in [9] (see Propositions 2.1.2 and 2.1.4 therein, respectively) that we will utilize in our further considerations.

Remark 2.5
The reader may easily convince himself that if G is a directed graph, then the following conditions are equivalent: Following these preliminary results, we can introduce a more general notion than a directed tree. It plays a pivotal role in our paper. Definition 2. 6 We say that a directed graph Note that directed trees fall within the scope of Definition 2.6, because for such graphs conditions (iii) and (iv) are satisfied by Proposition 2.3 and 2.4, respectively. In view of Remark 2.5 a relation between directed trees and directed semi-trees can be characterized as follows.
For completeness of exposition, let us discuss relations between conditions appearing in Definition 2.6.
Remark 2.8 Suppose that the conditions (i) and (ii) of Definition 2.6 are satisfied. If there exists v ∈ V • which has more then one parent, then, in general, none of conditions (iii) and (iv) holds. To see this, one may consider fairly easy examples of graphs such as (a) and (b), respectively, in Fig. 1. We also point out that neither (iii) implies (iv) nor (iii) follows from (iv). Indeed, it is enough to consider graphs (b) and (c), respectively, illustrated in Fig. 1.
As we see below, a directed semi-tree, similarly as a directed tree, cannot have more than one root.

Proposition 2.9
If F = (V, E) is a directed graph which is connected and satisfies the condition (iv) of Definition 2.6, then the set Root(G) contains at most one element. Proof Suppose that, contrary to our claim, there exist distinct u, v ∈ Root(F). By (iv), we can find w ∈ V such that u, v ∈ Des(w). Since Chi 0 ({w}) = {w}, it follows that at least one of these roots, say u, belongs to Chi n ({w}) for some n ≥ 1. This means that u has a parent, which is a contradiction.
We conclude this section with one more definition. A subgraph of a directed tree (resp. a directed semi-tree) F which itself is a directed tree (resp. directed semi-tree) is called a subtree (resp. a subsemi-tree) of F.
The proofs of the following propositions are left for the reader.
of F is a directed semi-tree if and only if F 1 is a connected graph and for all u, v ∈ V 1 , there exists w ∈ V 1 such that u, v ∈ Des(w).

The Bargmann Directed Semi-Trees
Here and in all that follows, we shall use the following notation. By N, Z + and C we mean the sets of all positive integers, non-negative integers, and complex numbers, respectively. The cardinality of N is denoted by ℵ 0 . Let We equip these sets with natural algebraic operations: Next, we set where δ i denotes a sequence from V n (resp. V ∞ ) whose the only non-zero entry is 1 at the i-th position. It is plain that B n := (V n , E n ), n ∈ N, and B ∞ := (V ∞ , E ∞ ) are directed graphs. We call them the n-finite Bargmann graph and the infinite Bargmann graph, respectively. Each B n can be regarded as a subgraph of B ∞ with the help of the natural identification An exemplary graph B n (for n = 3) is illustrated in Fig. 2.
In what follows, we confine our attention to B ∞ , because most of properties of all B n 's can be described in an analogous way. If some property is typical only for B ∞ , we will pay the reader's attention to this fact.
Let us discuss some properties of the Bargmann graph B ∞ .
as follows: (ii) Take distinct α, β ∈ V ∞ . If one of these elements equals , then the conclusion follows immediately from (i). Now suppose that α > and β > .
is an undirected path joining β and α, and so the graph B ∞ is connected. (iii) follows directly from (i), while (iv) is a consequence of (iii) and (2.1).
The next result is obvious, so we omit its proof.

Lemma 3.3
For all α ∈ V ∞ and β ∈ V • ∞ the following assertions hold: Remark 3.4 Note that in the case of B n (n ∈ N) assertion (ii) of Lemma 3.3 takes the form card(Chi(α)) = n. Lemma 3.5 For all α, β ∈ V ∞ such that α = β the following assertions hold: Let us briefly conclude our discussion in the form of the following theorem.
In view of Proposition 2.11 and the above theorem we can regard each B n (n ∈ N) as a directed subsemi-tree of B ∞ .

Segal-Bargmann Spaces as l 2 -Hilbert Spaces on Directed Semi-Trees
Let F = (V, E) be a directed semi-tree. We denote by l 2 (V ) the Hilbert space of all complex square-summable functions on V equipped with the standard inner product For u ∈ V , we define e u ∈ l 2 (V ) to be the characteristic function of the one- is a reproducing kernel Hilbert space [1], which is guaranteed by the reproducing property In what follows, we will need the ensuing simple observation.
For n ∈ N, we set for all holomorphic functions f, g : C n → C. Here ρ n denotes the Gaussian measure on C n defined by The Hilbert space (B n , ·, · n ) is called the Segal-Bargmann space of an n-th order [2]. Set It can be shown that {ε α } α∈Z n + is an orthonormal basis for B n [2].
we define the function f : C n → C as follows: is a unitary isomorphism, so we can identify the Segal-Bargmann space B n with the l 2 -Hilbert space defined on vertices of the Bargmann graph B n .
Next, we recall Bragmann's definition of the Hilbert space of an infinite order [3], which generalizes the definition of B n .
In all that follows, the set Z n + (resp. C n ) will be interpreted as a subset of V ∞ (resp. l 2 , where l 2 denotes the set of all square-summable complex sequences). Let we define the function f : l 2 → C as follows: The correctness of this definition as well as facts given below were discussed in [17]. Let Then the mapping J ∞ is isometric, and so (B ∞ , ·, · ∞ ) is a Hilbert space, which is called the Segal-Bargmann space of an infinite order. As a consequence, we can identify B ∞ with the l 2 -Hilbert space defined on vertices of the Bargmann graph B ∞ .

Remark 4.2
Recall that, in view of Sect. 3 we regard B n (n ∈ N) as a directed subsemi-tree of B ∞ . Hence, by Proposition 4.1, we treat l 2 (V n ) as a Hilbert subspace of l 2 (V ∞ ). Consequently, by this and the above identifications, B n can be regarded as a Hilbert subspace of B ∞ and the restriction of the unitary isomorphism J ∞ to l 2 (V n ) as J n . Note also that a Hilbert basis of B ∞ is the amalgamation of all bases of B n over all n ∈ N.
We shall return to Segal-Bargmann spaces in Sect. 6.

Properties of Weighted Shifts on Directed Semi-Trees
In all that follows, if A is an operator acting in a Hilbert space H, the symbol D(A) denotes its domain. The graph norm of A is defined as be an arbitrary subset of C. For v ∈ V • , we set In what follows, we also assume that for each v ∈ V • , The elements of λ are called weights. Note that if F is a directed tree, then λ v = {λ (par(v),v) } for v ∈ V • , and so (5.1) is trivially fulfilled. In [9], the symbol λ v was used to denote the unique element λ (par(v),v) .
For the set of weights λ, we define the operator S λ in l 2 (V ) as follows: where F is the mapping defined on functions f : V → C by We call the operator S λ a weighted shift on a directed semi-tree F. Let us show that the definition of S λ is well-posed. The case of a finite Par(v) is obvious, so let Par(v) = {u i : i ∈ N}. Then for n ≤ m (m, n ∈ N), by the Cauchy-Schwartz inequality, we get From (5.1) and the fact that f ∈ l 2 (V ) we deduce that which in connection with the above estimation justifies the correctness of the definition of S λ .
If S λ is a weighted shift on a directed tree, then the above definition of S λ coincides with that introduced in [9]. It should be stressed that, in view of Proposition 3.1.10 of [9], the most interesting results for weighted shifts on directed trees can be obtained in the case when card(V ) ≤ ℵ 0 . This motivated us to confine our considerations only to this case starting from Sect. 5.
It turns out that weighted shifts on directed semi-trees have many properties in common with those defined on directed trees. First, we prove the counterpart of Proposition 3.1.2 of [9]. For self-containedness, we include below its proof which goes in a similar fashion as in [9].

Proposition 5.1 A weighted shift S λ on a directed semi-tree F is a closed operator.
Proof Suppose that a sequence { f n } ∞ n=1 of elements from D(S λ ) is convergent to a function f ∈ l 2 (V ) and the sequence {S λ f n } ∞ n=1 converges to a function g ∈ l 2 (V ). Take v ∈ V . Then from the reproducing property [see (4.1)] we infer that Then, by the Cauchy-Schwartz inequality, we get which proves (5.4). Next, since (S λ f n )(root) = 0 by (5.2), we deduce from (5.3) that g(root) = 0. We conclude that , and so f ∈ D(S λ ) and g = S λ f . The proof is complete.
We now prove Theorem 5.2 which in combination with Theorem 7.4 may be regarded as an adaptation of Proposition 3.1.3 of [9] to the case of directed semitrees. It should however be pointed out that the last-mentioned result does not transfer automatically to our context and the proofs (in particular, that of Theorem 7.4) require some new methods. In what follows, we maintain the conventions of [9] that 0 ·∞ = 0 and that the sum over an empty set is equal zero. (u,v) : (u, v) ∈ E}. Then the following assertions hold:

Theorem 5.2 Let S λ be a weighted shift on a directed semi-tree F = (V, E) with weights λ = {λ
where · S λ denotes the graph norm of the operator S λ ; which gives (i).
(ii) For f ∈ D(S λ ), we obtain where (1) results from a combination of (5.1), the fact that f ∈ l 2 (V ) and the Cauchy-Schwartz inequality, whereas (2) is a consequence of the equality (iii) Take u ∈ V and observe that In view of (i), e u ∈ D(S λ ) if and only if the above sum is finite, which implies (a). Furthermore, take u ∈ V and observe that for w ∈ V • , we get If w = root, then again owing to (5.2) we get (S λ e u )(root) = 0, and ⎛ ⎝ v∈Chi(u) The latter sum is finite, because f ∈ D(S λ ) and (i) holds. Thus, again in view of (i), we deduce that f · χ W ∈ D(S λ ).
(v) is clear, because {e u } u∈V is an orthonormal basis of l 2 (V ).
For the sake of completeness, we state a simple observation.

Proposition 5.3 If λ is a set of weights and S λ is an operator in l 2 (V ) defined by the formula given in Theorem 5.2 (i), then S λ is a weighted shift on a directed semi-tree F.
Now let us turn to condition (v) of Theorem 5.2 and consider the following problem.

Problem 5.4 Does the inclusion {e
is a densely defined weighted shift on a directed semi-tree?
As condition (v) of Proposition 3.1.3 of [9] shows, the answer is in the affirmative in the case of weighted shifts on directed trees. We remain the above question in its full generality unanswered. Nevertheless, we can give a patrial solution to this problem. The inspiration for its proof comes from the last-mentioned result from [9].

Then the operator S λ is densely defined if and only if {e v } v∈V ⊂ D(S λ ).
Proof The only implication that requires care is (⇒). Let S λ be densely defined. Suppose on the contrary that there exists u ∈ V such that e u / ∈ D(S λ ). Then, by condition (iii a) of Theorem 5.2, we have For a fixed f ∈ D(S λ ), we get v∈Chi (u) λ (u,v) Indeed, by the Cauchy-Schwartz inequality, this sum is bounded above by and in view of (5.6) the latter expression is finite. A combination of (5.8), (5.7) and (5.9) leads to the conclusion that f, e u = f (u) = 0. Hence each f ∈ D(S λ ) is orthogonal to e u , but the operator S λ is densely defined, so we arrive at a contradiction.
It is worth mentioning that condition (5.6) is fulfilled for a directed semi-tree F = (V, E) such that card{v ∈ Chi(u) : card(Par(v)) > 1} < ℵ 0 for each u ∈ V.
A look at Example 5.7 given below reveals that (5.6) is not necessary for the inclusion {e v } v∈V ⊂ D(S λ ) to hold (consider u = v 0 therein). Next, note that if F is a directed tree, then the inequality in Theorem 5.2 (ii) turns into the equality taking a considerably simplified form as in Proposition 3.1.3 (ii) of [9]. That equality was used in [9] to show that where E V is the linear span of the set {e v : v ∈ V }, if the operator S λ is densely defined. In the case of weighted shifts on directed semi-trees we cannot employ an analogous argument to that of [9] to obtain (5.10). This is because the right-hand side of the inequality in Theorem 5.2 (ii) may be infinite if a directed semi-tree has a countable set of vertices, as evidenced in Example 5.7. It is also worth noting that the considered inequality may be strict even if its right-hand side is finite, which was shown in Example 5.8. We will return to the discussion on (5.10) in Sect. 7.
For m, n ∈ N, define weights λ as follows: A moment's thought reveals that, in view of condition (iii a) of Theorem 5.2, e v ∈ D(S λ ) for each v ∈ V . Now it suffices to take f ∈ D(S λ ) such that f (u) = 0 (e.g. f = e u ) to see that the right-hand side of the inequality in Theorem 5.2 (ii) is infinite. Remark additionally that, by (v) of Theorem 5.2, the operator S λ is densely defined.

Example 5.8 Let us consider
Let S λ be a weighted shift with weights λ such that whereas the right-hand side of (ii) takes the form Remark 5.9 It is also worth mentioning that for a weighted shift S λ on a directed tree F = (V, E) the description of D(S λ ) in Theorem 5.2 reduces to that given in [9, Proposition 3.1.3]. Indeed, then each v ∈ V • has a unique parent par(v). If we denote λ (par(v),v) by λ v , we get where the last equality results from Proposition 2.3.
We close this section by showing that the assumption that Par(v) ∩ W = ∅ or Par(v) ⊂ W for each v ∈ V in Theorem 5.2 (iv) cannot be dropped without affecting the validity of the result.

Generalized Creation Operators as Weighted Shifts on Directed Semi-Trees
In [3], Bargmann defined the generalized creation operator in B ∞ in a direction a ∈ l 2 by where ·, − is the standard inner product in l 2 . Such operators as well as, related with them, Bargmann's generalized anihilation operators (see [3]) were investigated in [17], where in particular was shown that each generalized creation operator in B ∞ is subnormal. An incorrect formula used in the proof of this result was recently improved and justified by the author in [22]. Our aim is to show that each operator A + a is unitarily isomorphic to a weighted shift on the infinite Bargmann directed semi-tree.
Recall that for each = (a 1 , a 2 , . . .) ∈ l 2 . Proposition 3 of [17], which plays a crucial role in our further consideration, says that for N), and . (6.4) Here and in all that follows, ε β is an element of the Hilbert basis in B ∞ given by (4.2), and β i stands for the i-th term of β ∈ V ∞ .
Next, define the operator A + a in l 2 (V ∞ ) as If a ∈ C n , then the domain and the image of A + a are included in B n , so A + a can be regarded as an operator acting in B n and A + a = J −1 n • A + a • J n (consult Remark 4.2). Note that then A + a acts in l 2 (V n ). In the rest of the paper, we confine our attention to the case when a ∈ l 2 , but our results can be easily reformulated for a ∈ C n . If it is necessary, we will highlight the difference between those two cases.
We are in a position to prove the main results of this section.

Theorem 6.2
For each a ∈ l 2 the operator A + a is a weighted shift on the infinite Bargmann semi-tree B ∞ .
Proof Take a = (a 1 , a 2 , . . .) ∈ l 2 . In view of (6.5), If this is the case, then the sum in (6.3) is finite. We can rewrite it, using on the way Lemma 3.3 (iii), as follows: and i(α, β) is a unique natural number such that α + δ i(α,β) = β, or equivalently α i(α,β) + 1 = β i(α,β) . Consequently, It also should be noted that because all but a finite number of terms β i equal zero.
Next, for ( f ) ∈ D(A + a ), using (6.4) and performing similar calculations as in (6.6), we get From (6.5), this and the fact that J −1 ∞ (ε β ) = e β for each β ∈ V ∞ we infer that Owing to the conditions (6.9), (6.8) and (6.10), we can call upon Proposition 5.3 to deduce that A + a is a weighted shift on B ∞ .
We know, by Remark 5.6, that {e α } α∈V n ⊂ D(A + a ) if a ∈ C n . It turns out that this result can be obtained in a more general case when a ∈ l 2 . However, to show this we cannot apply Proposition 5.5, because (as is easily checked) A + a does not satisfy condition (5.6).

Corollary 6.4
For each a ∈ l 2 the operator A + a is densely defined.
Finally, note that, since the operators A + a and A + a are unitarily isomorphic, the density of

A Core of a Weighted Shift on a Directed Semi-Tree
In this section we discuss another question which naturally arises from comparison of Proposition 3.1.3 of [9] with Theorem 5.2.
Problem 7.1 Does S λ = S λ | E V for a weighted shift S λ on a directed semi-tree, where E V is the linear span of the set {e v : v ∈ V }?
As already mentioned in Sect. 5, the answer is in the affirmative for weighted shifts on directed trees. In what follows, we shall formulate some sufficient conditions for the above equality to hold in a general case.
Let F = (V, E) be a directed semi-tree, where card(V ) ≤ ℵ 0 . Consider a sequence {S n } ∞ n=1 of subsets of V such that: (1) S n ⊂ S n+1 for each n ∈ N, card(Par(S n )) < ℵ 0 for each n ∈ N, It is easily seen that such a sequence {S n } ∞ n=1 can always be constructed for F.

Remark 7.2
The following relations between the above conditions can be observed. The implication (2) ⇒ (3) is not true in general, as evidenced in Example 7.3 (a). However, if we additionally assume that Chi(v) = ∅ for each v ∈ V , then (3) results from (2). Indeed, take v ∈ V and u ∈ Chi(v). In view of (2), u belongs to S n for some n ∈ N, which means that v ∈ Par(S n ). This proves (3). Next, note that the implication (3)⇒(2) does not hold even if conditions (1) and (4) are satisfied. To see this consult Example 7.3 (b). Further, it is clear that (1) implies that Par(S n ) ⊂ Par(S n+1 ) for each n ∈ N, but it can be shown that the converse statement is not true even if (2) and (3) hold. Finally, remark that all S n 's which satisfy (1)-(4) may be countable. Here we refer the reader to Example 7.3 (c). E) and S n 's be such that Fig. 3a). Then (2) holds, but (3) does not, because ∞ n=1 Par(S n ) = V \{v}. (b) Consider the graph F = (V, E) and S n 's such that Fig. 3b). It this case (3) holds. However, ,l) , v (k,l+1) ) : k, l ∈ Z + }.
For n ∈ N, we set The graph F with countable S n 's, illustrated in Fig. 3c, satisfies (1)-(4).
Before we move on, we need to introduce one more piece of notation. For W ⊂ V , we set Proof Take f ∈ D(S λ ) and set f n = w∈Par(S n ) f (w)e w for n ∈ N. If Par(S n ) = ∅, then f n = 0. Note that f n ∈ D(S λ ), because each set Par(S n ) is finite and (i) holds. Since S n ⊂ S n+1 , which yields Par(S n ) ⊂ Par(S n+1 ) for each n ∈ N, and V = ∞ n=1 Par(S n ), it follows that f n → f in l 2 (V ) as n → ∞. Next, observe that Take v ∈ V • . If v ∈ S • n , then Par(v) ⊂ Par(S • n ) ⊂ Par(S n ). If v / ∈ S • n and Par(v) ∩ Par(S n ) = ∅, then equivalently v ∈ (S n ) c [by (7.1)]. From this and (7.3) we infer that λ (u,v) Since S n ⊂ S n+1 for each n ∈ N and V • = ∞ n=1 S • n , it follows that B 1 ( f n ) → S λ f , while by the assumption (ii), B 2 ( f n ) → 0 as n → ∞. In view of (7.4), these facts imply that (S λ | E V ) f n → S λ f as n → ∞, which completes the proof.
Remark 7.5 It should be noted that the above argument can serve as an alternative proof of Theorem 3.1.3 (iv) of [9] for a class of weighted shifts acting on directed trees of a special kind. Indeed, let F = (V, E) be a directed tree with a root. For u ∈ V , we set l(u) = 0 if u = root, l(u) = 1 if (root, u) ∈ E, and l(u) = n (n ≥ 2) if there exists a finite sequence u 1 , . . . , u n−1 ∈ V such that (root, u 1 ), (u 1 , u 2 ), . . . , (u n−1 , u) ∈ E. Suppose additionally that F is such that card(S n ) < ℵ 0 for each n ∈ N, where S n := {u ∈ V : l(u) ≤ n}.
Then Par(S n ) = S n−1 and Chi(S n ) = S n+1 , so Chi(Par(S n )) = S n (n ∈ N). It is also clear that {S n } ∞ n=1 satisfies conditions (1)-(4). Consider a densely defined weighted shift S λ on F. In view of Theorem 3.1.3 (v) of [9], E V ⊂ D(S λ ). Next, note that condition (7.2) is trivially fulfilled, because S c n = ∅ (n ∈ N). As a result, we can apply Theorem 7.4 to deduce that S λ = S λ | E V .
We close this paper by showing that each operator A + a falls within the scope of Theorem 7.4.

Proposition 7.6
For each a ∈ l 2 , A + a = A + a | E V∞ , where E V ∞ is the linear span of the set {e α : α ∈ V ∞ }.