The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups

We prove a version of the Gohberg Lemma on compact Lie groups giving an estimate from below for the distance from a given operator to the set of compact operators. As a consequence, we obtain several results on bounds for the essential spectrum and a criterion for an operator to be compact. The conditions are given in terms of the matrix-valued symbols of operators.


Introduction
The original Gohberg Lemma was obtained during Gohberg's investigation of integral operators [Goh60], and its version on T 1 = {z ∈ C : |z| = 1} was obtained recently by [MW10], with application to the spectral properties of operators; see [Mol11,Pir11]. Related questions have been also considered on manifolds; see, e.g., [See65,Sch88] and related papers.
In this paper, we establish the Gohberg Lemma on general compact Lie groups, using the matrix quantization of operators developed in [RT10,RT12]. In particular, we give estimates for the distance from a given operator to the set of compact operators, as well as for the essential spectrum of the operator in terms of some quantities associated to the matrix symbols. The results contain the corresponding results obtained in [Mol11,Pir11] on T 1 .
Matrix-valued symbols have been quite useful in other studies of compactness of operators in cases when conditions on the kernel are less effective, for example, by providing criteria for operators to belong to Schatten classes [DR13b] and for nuclearity in L p -spaces [DR13a].
The structure of the paper is as follows. In Section 2, we briefly recall the necessary notions of Fourier analysis on compact Lie groups and of the matrix quantization of operators. In Section 3, we state our results. In Section 4, we prove the Gohberg Lemma (Theorem 3.1); and in Section 5, we give an application (Theorem 3.2).

Fourier analysis and matrix symbols on compact Lie groups
Let G be a compact Lie group and e its unit element. Let G be the unitary dual of G, i.e., the set of equivalence classes [ξ ] of the continuous irreducible unitary where the integral is (always) taken with respect to the Haar measure on G. The Fourier series becomes and Plancherel's identity takes the form We take (2.1) as the definition of the norm on the Hilbert space 2 ( G). Recall that f (ξ ) 2 HS = Tr( f (ξ ) f (ξ ) * ) is the Hilbert-Schmidt norm of the matrix f (ξ ). We define the matrix symbol of an operator T : where Tξ means T applied to the matrix components of ξ (x). It then follows that The correspondence between operators and symbols is one-to-one, and we write T σ for the operator given by (2.2) corresponding to the symbol σ(x, ξ). The quantization (2.2) has been studied extensively in [RT10,RT12], to which we refer for properties of the quantization and for the corresponding symbolic calculus.
Recall that the matrix components of ξ (x) are the eigenfunctions of the Laplacian (Casimir element) L on G corresponding to one eigenvalue, which we denote by λ 2 ξ , i.e., Lξ (x) i j = −λ 2 ξ ξ (x) i j for all 1 ≤ i, j ≤ d ξ . We write ξ : = (1 + λ 2 ξ ) 1/2 . Let 0 (G) be the usual class of operators that have symbols in Hörmander's class S 0 1,0 (R n ) in every local coordinate system. Let q 1 , . . . , q m ∈ C ∞ (G) satisfy q j (e) = 0, ∇q j (e) = 0 for all 1 ≤ j ≤ m, e is the only common zero of the family {q j } m j =1 , and rank{∇q 1 (e), · · · , ∇q m (e)} = dim G. We call such a collection of functions strongly admissible. We then define q j f (ξ ) := q j f (ξ ) and the difference operators α ξ : = α 1 q 1 · · · α m q m . We refer to [RT10], and especially to [RTW10], for the analysis of such difference operators.
It was proved in [RTW10] that T ∈ 0 (G) is equivalent to the condition that the matrix-valued symbol σ of T satisfies for all x ∈ G and [ξ ] ∈ G and for all α, β, where · op stands for the operator norm of the matrix multiplication. It was also shown in [RTW10] that the operator T ∈ 0 (G) is elliptic if and only if its matrix symbol σ(x, ξ) is invertible for all but finitely many [ξ ] ∈ G and for all such ξ ,

The Gohberg Lemma and applications
We formulate a version of the Gohberg Lemma first for operators in the Hörmander class 0 in order to relate it to well-known theory and for application in Theorem 3.2. Later, in Remark 4.2, we note that the result remains valid for a much more general class of operators.
Theorem 3.1 (Gohberg Lemma). Let T σ ∈ 0 (G) and σ(x, ξ) be the matrix symbol of T σ . Then for all compact operators K on L 2 (G), Note that d min is well-defined. Indeed, We note again that the condition T σ ∈ 0 (G) in Theorem 3.1 can be substantially relaxed; see Remark 4.2.
Before formulating our application of the Gohberg Lemma, let us first introduce some notation. Let X be a complex Banach space and A : X → X be a closed linear operator with dense domain D(A). The resolvent (A) of A is defined by Theorem 3.2. Let σ be the matrix symbol of a pseudo-differential operator T σ ∈ 0 (G), and let d max : = lim sup ξ →∞ {sup x∈G σ(x, ξ) op }. Then for T σ operating on L 2 (G), Moreover, if d max = 0, then T σ is a compact operator on L 2 (G).
In fact, T σ is compact if and only if d max = 0. Indeed, taking K = T σ in Theorem 3.1 shows that if d min = 0, then T σ is not compact.
In view of (3.2), d min ≤ d max . In case G = T n is the torus, d min = d max .

Proof of Theorem 3.1
First observe that by (3.2), d min is well-defined; and hence, for The definition of d min gives a sequence {x ξ n , ξ n } ∞ n =1 such that Therefore, so that u ξ n → 0 as ξ n → ∞ weakly. Hence, for a compact operator K , Ku ξ n L 2 (G) → 0 as ξ n → ∞.
Then, by compactness, for > 0 and sufficiently large n, where u is fixed and Ku ξ n L 2 (G) = G Ku ξ n (x) 2 HS dx 1/2 .
We postpone the proof of Lemma 4.1 and continue with the proof of Theorem 3.1.
Remark 4.2. With the same proof (and with N = 1), we have the following extension of the Gohberg Lemma without the assumption that the operator T σ belongs to 0 (G).
Let T σ : L 2 (G) → L 2 (G) be a bounded operator whose matrix symbol σ(x, ξ) satisfies for some ρ > 0 for all q ∈ C ∞ (G) with q(e) = 0 and all x ∈ G and [ξ ] ∈ G. Then the conclusion of the Gohberg Lemma in Theorem 3.1 remains true, namely, the estimate (3.1) holds for all compact operators K on L 2 (G).

Proof of Theorem 3.2
We first recall the following theorem, which gives another (equivalent) definition of Fredholm operators.

Theorem 5.1 (Atkinson's Theorem [Atk51]). Let A be a closed linear operator from a complex Banach space X into a complex Banach space Y with a dense domain D(A). Then A is Fredholm if and only if there exist a bounded linear operator B
: Y → X, a compact operator K 1 : X → X and a compact operator K 2 : Y → Y such that BA = I + K 1 on D(A) and AB = I + K 2 on Y . Then for ξ ≥ R, It follows from (2.4) that the operator T σ − λI : L 2 (G) → L 2 (G) is elliptic and hence a Fredholm operator; see, e.g., [Hör07,Section 19.5]. Thus We now turn to the proof of the last part of Theorem 3.2. Let L (L 2 (G)) and be the C * algebra of bounded linear operators on L 2 (G) and K (L 2 (G)) the ideal of compact operators on L 2 (G). Define multiplication on L (L 2 (G))/K (L 2 (G)) by A − K L (L 2 (G)) , [A] ∈ L L 2 (G))/K (L 2 (G) .
Using the Calkin algebra, we can reformulate (3.1) as [T σ ] C ≥ d min .
Assume now that d max = 0 and observe that T σ is compact if and only if [T σ ] = 0 in the Calkin algebra. Observe also that T σ is essentially normal on L 2 (G), i.e., T σ T * σ − T * σ T σ is compact. Indeed, T σ T * σ − T * σ T σ is an operator of order −1, so its compactness follows from the compactness of the embedding H 1 → L 2 . Consequently, [T σ ] is normal in L (L 2 (G))/K (L 2 (G)); and, therefore, r(T σ ) =