SUBSPACE GAPS AND WEYL’S THEOREM FOR AN ELEMENTARY OPERATOR

A range-kernal orthogonality property is established for the elementary operators ℰ(X)=∑i=1nAiXBi and ℰ*(X)=∑i=1nAi*XBi*, where A=(A1,A2,…,An) and B=(B1,B2,…,Bn) are n-tuples of mutually commuting scalar operators (in the sense of Dunford) in the algebra B(H) of operators on a Hilbert space H. It is proved that the operator ℰ satisfies Weyl's theorem in the case in which A and B are n-tuples of mutually commuting generalized scalar operators.


Introduction
For a Banach space operator T, T ∈ B(ᐄ), the kernel T −1 (0) and the range T(ᐄ) are said to have a k-gap for some real number k ≥ 1, denoted T −1 (0)⊥ k T(ᐄ), if [8, Definition, page 94].Recall from [10, page 93] that a subspace ᏹ of the Banach space ᐄ is orthogonal to a subspace ᏺ of ᐄ if m ≤ m + n for all m ∈ ᏹ and n ∈ ᏺ.This definition of orthogonality coincides with the usual definition of orthogonality in the case in which ᐄ = H is a Hilbert space.A 1-gap between T −1 (0) and T(ᐄ) corresponds to the range-kernel orthogonality for the operator T (see [1,2,8,14]).The following implications are straightforward to see where T(ᐄ) denotes the closure of T(ᐄ) and asc(T) denotes the ascent of T. A k-gap between T −1 (0) and T(ᐄ) does not imply that T(ᐄ) is closed, or even when T(ᐄ) is closed that ᐄ = T −1 (0) ⊕ T(ᐄ) (see, e.g., [1,2,23]).
This paper considers n-tuples A and B of mutually commuting scalar operators (in the sense of Dunford and Schwartz [10]) A i and B i , 1 ≤ i ≤ n, to prove that the operator Ᏹ µ := (Ᏹ − µI) ∈ B(B(H)) satisfies: (i) there exists a complex number λ = αexp iθ, α > 0 and 0 , where Ᏹ * λ = (Ᏹ * − λI).Furthermore, if the operators A i and B i in the n-tuples A and B are normal, then (ii) Ᏹ −1 λ (0) =Ᏹ −1 * λ (0).This compares with the fact that the operator Ᏹ may fail to satisfy the k-gap property of (i) or the Putnam-Fuglede-theorem-type commutativity property of (ii).However, if we restrict the length n of the n-tuples A and B to n = 1 (resp., n = 2), then both (i) and (ii) hold for all complex numbers λ [7,9] (resp., λ = 0 and λ = αexp iθ for some real number α > 0; see [7] and Theorem 2.4 infra).Our proof of (i) and (ii) makes explicit the relationship between the existence of a k-gap between the kernel and the range of the operator Ᏹ λ , and the Putnam-Fuglede commutativity property for n-tuples A and B consisting of mutually commuting normal operators.Letting the n-tuples A and B consist of mutually commuting generalized scalar operators (in the sense of Colojoarȃ and Foias ¸ [5]), it is proved that (i) a sufficient condition for Ᏹ λ (B(ᐄ)) to be closed is that the complex number λ is isolated in the spectrum of Ᏹ; (ii) f (Ᏹ) and f (Ᏹ * ) satisfy Weyl's theorem for every analytic function f defined on a neighborhood of the spectrum of Ᏹ, and the conjugate operator Ᏹ * satisfies a-Weyl's theorem.These results will be proved in Sections 2 and 3, but before that, we explain our notation and terminology.
The ascent of T ∈ B(ᐄ), asc(T), is the least nonnegative integer n such that T −n (0) = T −(n+1) (0) and the descent of T, dsc(T), is the least nonnegative integer n such that T n (ᐄ) = T n+1 (ᐄ).We say that T − λ is of finite ascent (resp., finite descent) if asc(T − λI) < ∞ (resp., dsc(T − λI) < ∞).The numerical range of T is the closed convex set of the set C of complex numbers (see [3]).A spectral operator (in the sense of Dunford) is an operator with a countable additive resolution of the identity defined on the Borel sets of C; a spectral operator T is said to be scalar type if it satisfies T = λE(dλ), where E is the resolution of the identity for T [11, page 1938 is the Fréchet algebra of all infinitely differentiable functions on C (endowed with its usual topology of uniform convergence on compact sets for the functions and their partial derivatives) and Z is the identity function on C (see [5,16]).We will denote the spectrum and the isolated points of the spectrum of T by σ(T) and isoσ(T), respectively.The closed unit disc in C will be denoted by D, and ∂D will denote the boundary of the unit disc D.
The operator of left multiplication by T (right multiplication by T) will be denoted by L T (resp., R T ).It is clear that [L S ,R T ] = 0 for all S,T ∈ B(ᐄ), where [L S ,R T ] denotes the commutator L S R T − R T L S .We will henceforth shorten (T − λI) to (T − λ).
An operator T ∈ B(ᐄ) is said to be Fredholm, T ∈ Φ(ᐄ), if T(ᐄ) is closed and both the deficiency indices α(T) = dim(T −1 (0)) and β(T) = dim(ᐄ/T(ᐄ)) are finite, and then the index of T, ind(T), is defined to be ind(T) = α(T) − β(T).The operator T is Weyl if it is Fredholm of index zero.The (Fredholm) essential spectrum σ e (T) and the Weyl spectrum σ w (T) of T are the sets Let π 0 (T) denote the set of Riesz points of T (i.e., the set of λ ∈ C such that T − λ is Fredholm of finite ascent and descent [4]), and let π 00 (T) denote the set of isolated eigenvalues of T of finite geometric multiplicity.Also, let π a0 (T) be the set of λ ∈ C such that λ is an isolated point of σ a (T) and 0 < dimker(T − λ) < ∞, where σ a (T) denotes the approximate point spectrum of the operator T. Clearly, π 0 (T) ⊆ π 00 (T) ⊆ π a0 (T).We say that Weyl's theorem holds for T if and a-Weyl's theorem holds for T if where σ ea (T) denotes the essential approximate point spectrum (i.e., σ ea (T) = ∩{σ a (T + K) : The concept of a-Weyl's theorem was introduced by Rakočević: a-Weyl's theorem for T implies Weyl's theorem for T, but the converse is generally false [20].
An operator T ∈ B(ᐄ) has the single-valued extension property (SVEP) at λ 0 ∈ C if for every open disc Ᏸ λ0 centered at λ 0 , the only analytic function f : Ᏸ λ0 → ᐄ which satisfies is the function f ≡ 0. Trivially, every operator T has SVEP at points of the resolvent ρ(T) = C \ σ(T); also T has SVEP at λ ∈ isoσ(T).We say that T has SVEP if it has SVEP at every λ ∈ C. The quasinilpotent part H 0 (T) and the analytic core K(T) of T ∈ B(ᐄ) are defined, respectively, by We note that H 0 (T − λ) and K(T − λ) are (generally) nonclosed hyperinvariant subspaces of T admits a generalized Kato decomposition, (GKD), if there exists a pair of T-invariant closed subspaces (ᏹ,ᏺ) such that ᐄ = ᏹ ⊕ ᏺ, the restriction T| ᏹ is quasinilpotent and T| ᏺ is semiregular.An operator T ∈ B(ᐄ) has a (GKD) at every λ ∈ isoσ(T), namely Fredholm operators are Kato type [13, Theorem 4], and operators T ∈ B(ᐄ) satisfying the following property: for some integer p ≥ 1, are Kato type at isolated points of σ(T) (but not every Kato type operator T satisfies property H(p)).

k-gap and the Putnam-Fuglede theorem
Let, as before, A = (A 1 ,A 2 ,...,A n ) and B = (B 1 ,B 2 ,...,B n ) be n-tuples of mutually commuting scalar operators, and let Ᏹ λ and Ᏹ * λ denote the elementary operators Ᏹ λ (X) The following theorem is the main result of this section.
Theorem 2.1.(i) There exists a real number α > 0 such that if Furthermore, if A and B are normally constituted, then , define the scalar α by α = √ α 1 α 2 , and define the operators C i and where I n denotes the identity of M n (C), it then follows that E is a contraction.Hence, In particular, 0 ∈ ∂W(B(B(H)),E 1 ).Notice that if 0 is an eigenvalue of Ᏹ α , then 0 is an eigenvalue of E 1 .It follows from Sinclair [23, Proposition 1] that for all X ∈ E −1 1 (0) and Y ∈ B(H).In particular, asc(E 1 ) ≤ 1, which by a result of Shulman [21] implies that where it follows that E * is a contraction and 0 is an eigenvalue of E * 1 in ∂W(B(B(H)),E * 1 ).Hence, . The proof of (ii) is now a consequence of the observation that To prove (i), we let . This completes the proof of the theorem.
The following corollary is an immediate consequence of the fact that asc(Ᏹ α ) ≤ 1.

and all Y ∈ B(H).
Furthermore, if A and B are normally constituted, then for all λ as in part (i).Let the Hilbert space H be separable, and let Ꮿ p denote the von Neumann-Schatten p-class, 1 ≤ p < ∞, with norm • p .Then Theorem 2.3 has the following Ꮿ p version.

Theorem 2.4. (i) There exists a complex number λ
Furthermore, if A and B are normally constituted, then (ii) Ᏹ −1 λ (0) ∩ Ꮿ p = Ᏹ −1 * λ (0) ∩ Ꮿ p for all λ as in part (i).Proof.Define the real numbers α i , i = 1,2, as in the proof of Theorem 2.1, define the normal operators C i and D i by ) is a contraction.Now argue as in the proof of Theorem 2.1.
As we will see in the following section, H 0 (Ᏹ λ ) = Ᏹ −p λ (0) for all λ ∈ C and some integer p ≥ 1 (i.e., Ᏹ λ satisfies property H(p)), which implies that asc(Ᏹ λ ) ≥ 1 for all λ ∈ C. (Here, as also elsewhere, the statement asc(T) ≥ 1 is to be taken to subsume the hypothesis that T is not injective.)However, if the n-tuples A and B are of length n = 1, then asc(Ᏹ λ ) ≤ 1 for all λ ∈ C and for a number of classes of not necessarily scalar or normal operators A 1 and B 1 (see [7,9]).If n = 2 and B 1 = A 2 = I, then asc(Ᏹ λ ) ≤ 1 (once again for A 1 and B 2 belonging to a number of classes of operators more general than the class of scalar operators [7]).Again, if n = 2, then asc(Ᏹ λ ) ≤ 1 for λ = 0 and λ = αexp iθ, as follows from Theorem 2.1 and the following argument.Define the normal operators M i and N i , i = 1,2, as in the proof of Theorem 2.1.[14] or [8]), which implies that asc(φ) = asc(Ᏹ) ≤ 1.The following corollary, which generalizes [14, Theorem 2], is now obvious.
number λ is as in Theorem 2.3, then asc(Ᏹ µ ) ≤ 1, and Ᏹ −1 µ (0)⊥ k Ᏹ µ (B(H)) for µ = 0,λ.Furthermore, if A and B are normally constituted, then Ᏹ −1 µ (0) = Ᏹ −1 * µ (0) for µ = 0,λ.Perturbation by quasinilpotents.Recall that every spectral operator T ∈ B(ᐄ) is the sum T = S + Q of a scalar type operator S and a quasinilpotent operator Q such that [S,Q] = 0 [11].Let A = (J 1 ,J 2 ) and B = (K 1 ,K 2 ) be tuples of operators in B(H) such that , where Ᏹ(X) is defined as in Corollary 2.5 and φ(X Recall that the sum of two commuting quasinilpotent operators, as well as the product of two commuting operators one of which is quasinilpotent, is quasinilpotent [5, Lemma 3.8, Chapter 4].Representing the operator X → SXT by X → L S R T (X), where (S,T) denotes any of the operator pairs

and assuming that the operators in the sets
mutually commute, it follows that the operator φ is quasinilpotent.

Weyl's theorem
If ] = 0 for all 1 ≤ i, j ≤ n, the mutual commutativity of the n-tuples implies that [L Ai R Bi , L Aj R Bj ] = 0 for all 1 ≤ i, j ≤ n, the generalized scalar operators L Ai R Bi and L Aj R Bj have two commuting spectral distributions, and (hence that) L Ai R Bi + L Aj R Bj is a generalized scalar operator.A finitely repeated application of this argument implies that E λ is a generalized scalar operator for all λ ∈ C. Thus for some integer p ≥ 1 and all λ ∈ C see [5,Theorem 4.5,Chapter 4].In particular, asc(E λ ) ≤ p < ∞ for all λ ∈ C and E(= E 0 ) has SVEP.
(b) The following implications hold:
It is evident from Proposition 3.1(a) that a sufficient condition for E λ to have closed range is that λ ∈ isoσ(E).Proposition 3.1(b) implies that both E and E * satisfy Weyl's theorem: more is true.Let H(σ(E)) denote the set of functions f which are defined and analytic on an open neighborhood of σ(E).
A,B ∈ B(ᐄ) are generalized scalar operators, then L A ,R B ∈ B(B(ᐄ)) are commuting generalized scalar operators with two commuting spectral distributions, which implies that L A R B and L A + R B are generalized scalar operators (see [5, Theorem 3.3, Proposition 4.2, Theorem 4.3, Chapter 4]).Let A = (A 1 ,A 2 ,...,A n ) and B = (B 1 ,B 2 ,...,B n ) be ntuples of mutually commuting generalized scalar operators in B(ᐄ), and let the elementary operator E λ ∈ B(B(ᐄ)) be defined by E λ (X) = n i=1 A i XB i − λX.Since [L Ai ,R Bj