Upper triangular operator matrices, asymptotic intertwining and Browder, Weyl theorems
© Duggal et al.; licensee Springer 2013
Received: 8 February 2013
Accepted: 7 May 2013
Published: 29 May 2013
Given a Banach space , let denote the upper triangular operator matrix , and let denote the generalized derivation . If , then , where stands for the spectrum or a distinguished part thereof (but not the point spectrum); furthermore, if is a Riesz operator which commutes with , then , where stands for the Fredholm essential spectrum or a distinguished part thereof. These results are applied to prove the equivalence of Browder’s (a-Browder’s) theorem for , , and . Sufficient conditions for the equivalence of Weyl’s (a-Weyl’s) theorem are also considered.
MSC:47B40, 47A10, 47B47, 47A11.
A Banach space operator , the algebra of bounded linear transformations from a Banach space into itself, satisfies Browder’s theorem if the Browder spectrum of T coincides with the Weyl spectrum of T; T satisfies Weyl’s theorem if the complement of in is the set of finite multiplicity isolated eigenvalues of T. Weyl’s theorem implies Browder’s theorem, but the converse is generally false (see [1–3]). Let and denote, respectively, the upper triangular operators and for some operators . It is well known that for , and . The problem of finding sufficient conditions ensuring the equality of the spectrum (and certain of its distinguished parts) of and , along with the problem of finding sufficient conditions for satisfies Browder’s theorem and/or Weyl’s theorem to imply satisfies Browder’s theorem and/or Weyl’s theorem (and vice versa), has been considered by a number of authors in the recent past (see , and some of the references cited there). For example, if either or B has the single-valued extension property, SVEP for short, then . Again, if , then [, Proposition 3.2] and satisfies Browder’s theorem if and only if satisfies Browder’s theorem [, Theorem 4.8]; furthermore, in such a case, satisfies Weyl’s theorem if and only if satisfies Weyl’s theorem if and only if [, Theorem 5.1]. The equality may be achieved in a number of ways: if either A and , or A and B, or and , or B and have SVEP, then [, Proposition 4.5]. In this paper we consider conditions of another kind, conditions which do not assume SVEP.
Given , S and T are said to be asymptotically intertwined by if . Here is the generalized derivation and . Evidently, S and T asymptotically intertwined by X does not imply T and S asymptotically intertwined by X. Furthermore, S and T asymptotically intertwined by X does not imply , not even ; see [, Example 3.5.9]. However, as we shall see, if A, B, C are as in the definition of above, then A and B asymptotically intertwined by C implies the equality of the spectra, and many distinguished parts thereof to spectrum of and . We prove in the following that if , then satisfies Browder’s theorem if and only if satisfies Browder’s theorem. If, additionally, the isolated points of are poles of the resolvent of , then satisfies Weyl’s theorem if and only if satisfies Weyl’s theorem. Extensions to a-Browder’s theorem, a-Weyl’s theorem and perturbations by Riesz operators are considered.
2 Notation and complementary results
are hyper-invariant (generally non-closed) subspaces of S such that for every integer and . Recall also that if , then .
We say that S has the single valued extension property, or SVEP, at if for every open neighborhood U of λ, the only analytic solution f to the equation for all is the constant function ; we say that S has SVEP if S has a SVEP at every . It is well known that finite ascent implies SVEP; also, an operator has SVEP at every isolated point of its spectrum (as well as at every isolated point of its approximate point spectrum).
satisfies Browder’s theorem, shortened to S satisfies Bt, if (if and only if , see [, p.156]); S satisfies Weyl’s theorem, shortened to S satisfies Wt, if (if and only if S satisfies Bt and ) [, p.177]. The implication is well known.
An isolated point is a pole (of the resolvent) of if . In such a case we say that S is polar at λ; we say that S is polaroid (resp., polaroid on a subset F of the set of isolated points of ) if S is polar at every (resp., at every ). Let denote the set of poles of S.
Throughout the following, shall denote the diagonal operator and shall denote the upper triangular operator matrix , for some operators . Recall, [, Exercise 7, p.293], that and .
Lemma 2.1 If , then .
i.e., is polar at λ. Conversely, if , then and implies and are both finite, hence equal. Thus is polar at λ. □
Remark 2.2 A number of conditions guaranteeing (the spectral equality) are to be found in the literature. Thus, for example, if or B has SVEP, or if , or [, (I) p.5 and Proposition 3.2], then . Compact operators have SVEP; hence, if either of A or B is compact, then .
Lemma 2.1 shows that if B is a compact operator then . A proof of the following lemma may be obtained from that of Lemma 2.1: we give here an independent proof, exploiting the additional information contained in the hypothesis.
Lemma 2.3 If , then .
Proof Once again we consider points . Let . Then implies , and this in turn implies and . Since λ is isolated in and , [, Theorem 3.77]. Consequently, ; furthermore, since , . Conversely, if , then and , and hence (since λ is isolated in and ) . This, as above, implies . □
The following technical lemma will be required in the sequel.
Lemma 2.4 If A is polaroid on and , then .
either or . If , then (since ) contains an orthonormal sequence such that for all . But then , a contradiction. Hence . Since and A is (by hypothesis) polar at λ (with, as observed above, ) . Thus , and so there exists a sequence such that for all . But then for all , and hence . This contradiction implies that we must have . Since , we conclude that . □
Let denote the generalized derivation , and define by . The operators are said to be asymptotically intertwined by the identity operator if ; S, T are said to be quasi-nilpotent equivalent if [, p.253]. Quasi-nilpotent equivalence preserves a number of spectral properties [, Proposition 3.4.11]. In particular:
Lemma 2.5 Quasi-nilpotent equivalent operators have the same spectrum, the same approximate point spectrum and the same surjectivity spectrum.
S is upper semi-Fredholm, , if and only if is injective, if and only if is bounded below;
S is lower semi-Fredholm, , if and only if is surjective;
S is Fredholm, , if and only if is invertible.
Lemma 3.1 For every , , and .
The following theorem is essentially known  we provide here an alternative proof, using quasi-nilpotent equivalence and the construction above. Let denote either of , , , , , , , and .
Theorem 3.2 Let . If R is a Riesz operator which commutes with S, then , where .
To complete the proof, we prove next that ; the proof for and is similar, and left to the reader. It would suffice to prove that . Suppose that . Then (and ), hence for all . For an operator T, let and denote, respectively, the closure of the hyper kernel and the hyper range of T. Then is constant on , and so, since , it follows that . Consequently, has SVEP at 0 [, Corollary 2.26]. But then since , is Browder. Considering proves . □
The following lemma appears in [, Lemma 2.3]. Let . Clearly, .
Lemma 3.3 If , and R is a Riesz operator which commutes with S, then .
Theorem 3.4 If , then , where .
[, Corollary 3.23, Theorem 3.23 and Theorem 3.27]. Hence , where . □
Remark 3.5 If is the operator such that the entries A, B, C and D mutually commute, then [, Theorem 2.3], where . Dispensing with the mutual commutativity hypothesis and assuming instead that , C commutes with A and B, and , an argument similar to that used to prove Theorem 3.4 shows that , where .
satisfies Bt if and only if satisfies Bt.
Let , , be Riesz operators such that commutes with . Then satisfies satisfies satisfies satisfies Bt.
- (a)Recall that an operator S satisfies Bt if and only if . Hence the following implications hold:
- (b)The hypothesis implies that and are quasi-nilpotent equivalent (⟹ by Theorem 3.4 that , where ). The operator R being Riesz, Theorem 3.2 implies , where or and or . The (two way) implications
now complete the proof. □
The equivalence S satisfies satisfies Bt is well known. This does not hold for a-Bt: S satisfies a-Bt does not imply satisfies a-Bt (or vice versa). We say that S satisfies s-Bt if satisfies a-Bt (equivalently, if ). It is easily seen, we leave the verification to the reader, if either or satisfies s-Bt, then (in Theorem 3.6) , , and all satisfy s-Bt.
We consider next a sufficient condition for the equivalence of Weyl’s theorem for operators and such that . We say in the following that an operator S is finitely polaroid on a subset if every is a finite rank pole of S. Evidently, is finitely polaroid if and only if A and B are finitely polaroid.
If A is polaroid, then satisfies Wt if and only if satisfies Wt.
Let , , be Riesz operators such that commutes with . A sufficient condition for the equivalence satisfies satisfies Wt is that is finitely polaroid.
- (b)Start by observing that , and hence is finitely polaroid if and only if is finitely polaroid (Lemma 2.3). Suppose satisfies Wt. Then the implication combined with Theorem 3.6(b) implies that both and satisfy Bt. As noted in the proof of Theorem 3.6(b), , or . Furthermore, since and are quasi-nilpotent equivalent, , (Theorem 3.4). Hence
If and , then is invertible and so . Hence, since , . If, instead, , then (Lemma 3.3) (since is finitely polaroid) (Lemma 2.3) , and this as above implies . Hence , and satisfies Wt. The converse, satisfies satisfies Wt follows from a similar argument (recall that is finitely polaroid follows from the hypothesis that is finitely polaroid). □
This is seen as follows. The implication satisfies satisfies Bt and satisfies satisfies Bt are clear from Theorem 3.6(b). If satisfies Bt, then the hypothesis is finitely polaroid implies satisfies Wt. By Theorem 3.6(b), satisfies Bt, i.e., . Let . If , then () (since ); if , then (by Lemma 3.3) and so (since is finitely polaroid) . Thus, in either case, , and hence satisfies Wt. The proof for satisfies satisfies Wt is similar: recall from Lemma 2.3 that finitely polaroid implies finitely polaroid.
a-Wt. satisfies a-Weyl’s theorem, a-Wt for short, if T satisfies a-Bt and (equivalently, if ), where . We say in the following that T is a-polaroid if T is polar at every . Trivially, a-polaroid implies polaroid (indeed, in such a case), but the converse is not true in general. Theorem 3.8 has an a-Wt analogue, which we prove below. We note, however, that the perturbation of an operator by a commuting Riesz operator preserves neither its spectrum nor its approximate point spectrum: this will, per se, force us into making an assumption on the approximate point spectrum of and in the analogue of Theorem 3.8(b).
If is a-polaroid, then satisfies a-Wt if and only if satisfies a-Wt.
Let , , be Riesz operators such that commutes with . If , then a sufficient condition for the equivalence satisfies satisfies a-Wt is that is finitely a-polaroid.
- (b)If , then it follows from Lemma 2.4 and Theorem 3.4 that
whenever either of , , and satisfies a-Bt. Observe that the hypothesis is finitely a-polaroid implies ; hence (since ) for every choice of . We prove now that if either of and satisfies a-Wt, then : this would then imply that if one satisfies a-Wt, then so does the other.
Suppose satisfies a-Wt. Then () and . Let ; then implies . Thus . Consequently, in this case. Suppose next that satisfies a-Wt. Then and . Let ; then implies . As above, this implies . □
The following corollary is immediate from Theorem 3.10(b).
Corollary 3.11 Suppose that . If , , are quasi-nilpotent operators such that commutes with , then a sufficient condition for the equivalence satisfies satisfies a-Wt is that is finitely a-polaroid.
Work carried out together whilst the first author was visiting Korea.
This work was supported by the Incheon National University Research Grant in 2012.
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