View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 289 (2004) 419–430 www.elsevier.com/locate/jmaa On essential spectra of operator-matrices and their Feshbach maps J. Lutgen Fachbereich Mathematik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany Received 25 November 2002 Submitted by F. Gesztesy Abstract A connection between the essential spectrum of certain operator-matrices and essential spectra of the corresponding “Feshbach maps” is discussed and applied to some concrete rational operator- valued functions. 2003 Elsevier Inc. All rights reserved. 1. Introduction The primary goal in this paper is to make sense of and to characterize the essential spectrum of certain operator-valued functions λ → (A − λ) − B(D − λ)−1C (λ ∈ Ω ⊂ C) involving the “spectral parameter” λ rationally. The first observation is that operator- functions of this form arise as so-called Feshbach maps (cf. [3,5]) corresponding to certain operator-matrices. In other contexts these maps are often referred to as Livsic matrices (cf. [9]) or as transfer functions (cf. [16]). If a linear operator H : X → X in a com- plex Banach space X is written with respect to a preferred topological decomposition X = X1 ⊕ X−1 as an operator-matrix 1H11H−1 H = : X1 ⊕ X−1 → X1 ⊕ X−1, (1) −1H1 −1H−1 E-mail address: [email protected]. 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.08.025 420 J. Lutgen / J. Math. Anal. Appl. 289 (2004) 419–430 where iHj := Pi HPj and Pj is the continuous projection of X onto Xj induced by the decomposition, then the Feshbach maps corresponding to H are the operator-valued func- tions −1 λ → Fj (λ) := (j Hj − λ) − j H−j (−j H−j − λI) −j Hj (j =±1), (2) whose values make sense as linear operators in Xj for λ ∈ ρp(−j H−j ), where we use the notation ρp(·) := C \ σp(·) for the complement of the point spectrum. The point spectrum σp(Fj ) and the spectrum σ(Fj ) of an operator-function Fj are defined as usual (cf. [15]): σp(Fj ) is the set of all λ for which Fj (λ) is defined and not injective and σ(Fj ) is the set of all λ for which Fj (λ) is defined, closed and does not have a bounded (everywhere defined) inverse. The eigenspace corresponding to an eigenvalue λ is just Ker(Fj (λ)).Itis well known and easy to prove from the definitions that σp(H) ∩ ρp(j Hj ) = σp(F−j ) ∩ ρp(j Hj )(j=±1), (3) the eigenspaces of Fj and H corresponding to a common eigenvalue are isomorphic (cf. [3] or Lemma 3.2 in [1]) and, under some mild assumptions, chains of eigen- and associated vectors of Fj for an eigenvalue λ correspond to Jordan chains for H at λ (cf. [12]). Further- more, it is a well-known fact that under some boundedness assumptions formula (3) holds for spectra, i.e., with the subscript “p” dropped (cf. [3,5]). This fact is also apparent from −1 the factorization in the proof of Theorem 3 under the hypotheses specified there. Fj (λ) is also called the compressed resolvent of H to Xj (cf. [9]), since under suitable conditions −1 it equals Pj (H − λ) Pj in Xj . Assuming Fj (λ) is closable for each λ we propose to define the essential spectrum of the operator-function λ → Fj (λ), in analogy to the point and whole spectrum above, as the set of λ such that Fj (λ) is not a Fredholm operator and we denote this set by σe(Fj (·)). In Section 3 this proposal is considered closely: There Theorem 3 says essentially that this “naive” definition is indeed the correct one; in particular, under suitable hypotheses, H is closable if and only if F−j (λ) is closable for some, or equivalently, for all λ ∈ ρB (j Hj ) and in this case we have ¯ ¯ σe(H) ∩ ρB (j Hj ) = σe(F−j ) ∩ ρB (j Hj )(j=±1). (4) Here ρB (·) is the so-called “Browder resolvent set,” which is discussed in Section 2. The Feshbach maps F−j extend naturally to operator-functions on ρB (j Hj ) by replacing the resolvent by the Browder resolvent. In general ρB (·) is smaller than the “Fredholm resol- vent set” ρe(·) := C \ σe(·), but for typical operators j Hj arising in applications, e.g., for normal or compact operators or operators with compact resolvent, the two sets are equal, ρB (j Hj ) = ρe(j Hj ). Section 2 contains technical results needed in the proof of Theorem 3 which is based on the well-known “lower-upper factorization” for 2 × 2 operator-matrices commonly used to investigate spectral and Fredholm properties of bounded operator-matrices (cf. [8]). This factorization has also been applied in [2,17] to unbounded operator-matrices. In [2] the authors assume that 1H1 has compact resolvent, that the Feshbach map F−1(λ) + λ in the lower component space may be written as a fixed operator S0 plus a compact, λ-dependent perturbation and that some additional boundedness and compactness hypotheses hold. They show that the essential spectrum of the closure of the resulting operator-matrix is J. Lutgen / J. Math. Anal. Appl. 289 (2004) 419–430 421 σe(S0). In [17] the method is extended to show that even when the upper left entry 1H1 does not have a compact resolvent the essential spectrum of the operator matrix can still be obtained as the union of the essential spectra of the operators 1H1 and F−1(λ) + λ for any λ ∈ (1H1). So the basis of Theorem 3 is similar, but the approach is different in that, as a first step, we only try to connect Fredholm properties of the operator-matrix with Fredholm properties of the Feshbach maps. This and the use of the Browder resolvent allows us to formulate the result more generally, relax the assumptions somewhat and make the proof more compact. In Sections 4 and 5 the essential spectrum of AM H = s : L2(I) ⊕ L2(I) → L2(I) ⊕ L2(I) (5) Mt Mu is investigated, where I = (a, b) (−∞ a<b ∞), Ms , Mt and Mu are multiplication operators in L2(I) generated by complex-valued, measurable functions s, t and u and A is 2 a selfadjoint extension in L (I) of the minimal operator generated by a real, formally := n − n−j { (n−j) }(n−j) selfadjoint quasi-differential expression τf(x) j=0( 1) pj (x)f (x) ∈ 1 of order 2n with the usual local integrability assumptions 1/p0,pj Lloc(I) (1 j n) (cf. [6,19]). Furthermore, we assume throughout that the coefficient p0 is positive definite, the coefficients p1,...,pn are bounded from below, ess ran(u) = C, i.e., ρ(Mu) =∅,and ∈ 2 + ∞ s,t L0(I) L (I), i.e., each of s and t can be expressed as a sum of a square integrable function which converges to zero at infinity and a bounded function. The main result of the paper (Theorem 8) says, under some additional hypotheses, that H in (5) is closable, the essential spectrum of its closure is σe(A) ∪ ess ran(u) and the Fredholm index of H¯ − µ is zero for µ outside the essential spectrum. This result is obtained by combining an application of Theorem 3 to the concrete case (5) (Section 4) with a special decomposition principle (Section 5). One way of apply- ing the abstract Theorem 3 to obtain information about the essential spectrum of concrete operator matrices is to show that F−j (µ) = (M − µ) + K(µ),whereM is a fixed (closed) operator and K(µ) is M-compact for each µ (so that F−j (µ) is Fredholm if and only if −1 M − µ is). Here F−1(µ), for example, has the form (Mu − µ) − −1H1(A − µ) 1H−1. For example, in the concrete problem considered in [2], A, −1H1 and 1H−1 are regular differential operators where the sum of the orders of the off-diagonal operators equals the −1 order of A, so that the term −1H1(A − µ) 1H−1 is of order zero and can be written formally as a bounded multiplication operator independent of µ plus a compact integral operator and one obtains the desired form for the Feshbach map. In the singular case, as here, this approach is more technical and requires a lot information about kernel functions, respectively, about solutions of corresponding differential equations and we use a different −1 approach to obtain relative compactness of the term −1H1(A − µ) 1H−1 in Section 4. H We remark that the main result cannot be obtained by considering to be a perturbation A 0 0 Ms of by except in the trivial case when Ms and Mt are the zero operators. 0 Mu Mt 0 For example, a bounded off-diagonal part is compact relative to the diagonal part if and −1 −1 2 only if Ms (Mu − λ) and Mt (A − λ) are compact in L (I) (λ ∈ ρ(A)∩ ρ(Mu)). Since the first product is just the multiplication operator in L2(I) defined by the function x → s(x)(u(x) − λ)−1, it can only be compact if this is the zero function, i.e., only if s = 0. Arguing with adjoints shows similarly that t = 0. 422 J. Lutgen / J. Math. Anal. Appl. 289 (2004) 419–430 The spectral problem for the operator-function F1 corresponding to (5) is then just s(x)t(x) Af (x ) − f(x)= λf(x), f ∈ L2(I), (6) u(x) − λ which is the subject of a series of investigations in the literature (cf.