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

In this paper 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 on compact Lie groups. As a consequence, we prove 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
In this paper we establish a version of the Gohberg lemma in the setting of compact Lie groups and apply it to study the compactness of pseudo-differential operators and give bounds for their essential spectrum. The original Gohberg lemma has been obtained by Gohberg [Goh60] in the investigation of integral operators, and its version on the unit circle T 1 has been recently obtained by [MW10], with application to the spectral properties of operators, see [Mol11,Pir11]. 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 the unit circle. The 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, see [DR13b], and criteria for nuclearity in L p -spaces, see [DR13a].
In Section 2 we briefly recall the necessary notions of the 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 given in Theorem 3.1, and in Section 5 we prove an application of the Gohberg lemma given in Theorem 3.2.

Fourier analysis and matrix symbols on compact Lie groups
Let G be a compact Lie group with the unit element e, and let G be its unitary dual, consisting of the equivalence classes [ξ] of the continuous irreducible unitary For a function f ∈ C ∞ (G) we can define its Fourier coefficient at ξ by where the integral is (always) taken with respect to the Haar measure on G. The Fourier series becomes with the Plancherel's identity taking the form which we take as the definition of the norm on the Hilbert space ℓ 2 ( G), and where f (ξ) 2 HS = Tr( f (ξ) f (ξ) * ) is the Hilbert-Schmidt norm of the matrix f (ξ). Given an operator T : where T ξ means that we apply T to the matrix components of ξ(x). In this case we can prove that The correspondence between operators and symbols is one-to-one, and we will write T σ for the operator given by (2.2) corresponding to the symbol σ(x, ξ). The quantization (2.2) has been extensively studied in [RT10,RT12], to which we refer for its properties and for the corresponding symbolic calculus. We note 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. we have Lξ(x) ij = −λ 2 ξ ξ(x) ij for all 1 ≤ i, j ≤ d ξ . We denote ξ := (1 + λ 2 ξ ) 1/2 . We now briefly describe the class Ψ 0 (G) of Hörmander's pseudo-differential operators on G in terms of the matrix symbols. Here, Ψ 0 (G) stands for the usual class of operators that have symbols in Hörmander's class S 0 1,0 (R n ) in every local coordinate system.
It was proved in [RTW10] that T ∈ Ψ 0 (G) is equivalent to the condition that its matrix-valued symbol σ satisfies for all x ∈ G and [ξ] ∈ G, and for all α, β, where · op stands for the operator norm of the matrix multiplication. The difference operators ∆ α ξ in (2.3) are defined as follows. Let q 1 , . . . , q m ∈ C ∞ (G) be such that q j (e) = 0, ∇q j (e) = 0, for all 1 ≤ j ≤ m, the unit element e is the only common zero of the family {q j } m j=1 , and such that rank{∇q 1 (e), · · · , ∇q m (e)} = dim G. We call such a collection strongly admissible. Then we set ∆ q j f (ξ) := q j f (ξ) and ∆ α ξ := ∆ α 1 q 1 · · · ∆ αm qm . We refer to [RT10] and especially to [RTW10] for the analysis of such difference operators.
It was shown in [RTW10] that the operator T ∈ Ψ 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 we have

Gohberg lemma and applications
We define σ(x, ξ)σ(x, ξ) * min to be the smallest eigenvalue of the positive matrix σ(x, ξ)σ(x, ξ) * , that is, if λ 1 (x, ξ), λ 2 (x, ξ), . . . , λ d ξ (x, ξ) ≥ 0 are the eigenvalues of σ(x, ξ)σ(x, ξ) * then we set We formulate a version of the Gohberg Lemma first for operators in the Hörmander class Ψ 0 to relate it with the well-known theory and to be used in the application in Theorem 3.2, but later, in Remark 4.2, we note that the result remains valid for a much more general class of operators.
Then for all compact operators K on L 2 (G), we have We note that d min is well-defined. Indeed, we have We note again that the condition T σ ∈ Ψ 0 (G) in Theorem 3.1 can be substantially relaxed, see Remark 4.2.
To formulate an application of the Gohberg lemma, let us first introduce some notation. Let A : X → X be a closed linear operator with dense domain D(A) in the complex Banach space X. Theorem 3.2. Let σ be the matrix symbol of a pseudo-differential operator T σ ∈ Ψ 0 (G). Let Then for T σ on L 2 (G) we have Moreover, if d max = 0, then T σ is a compact operator on L 2 (G).
We observe that it follows from Theorem 3.1 that if d min = 0, then T σ is not compact, since otherwise we could take K = T σ . In other words, if T σ is compact, then d min = 0. From this point of view, the converse to this is given by the last statement of Theorem 3.2.
We note that we always have d min ≤ d max in view of (3.2). If G = T n is the torus, we have d min = d max .

Proof of Theorem 3.1
This section is devoted to the proof of Theorem 3.1.
First we observe that by (3.2), d min is well-defined, and hence for every From the definition of d min there exists a sequence (x ξn , ξ n ) such that ξ n → ∞ and Therefore, we have the equality weakly. Hence for a compact operator K we have Ku ξn L 2 (G) → 0 as ξ n → ∞. Then for any ǫ > 0 and sufficiently large n we have by compactness where u is fixed and Ku ξn L 2 (G) = G Ku ξn (x) 2 HS dx 1/2 . We now define T σ u ξn := (T σ (u ξn ) ij ) 1≤i,j≤d ξn ∈ C d ξn ×d ξn by T σ acting on the components of the matrix-valued function u ξn .
We postpone the proof of Lemma 4.1 and continue with the proof of Theorem 3.1.
Let us fix u ∈ C ∞ (G) such that u = 0. Then for any ǫ > 0 there exists N(u) such that for any n ≥ N(u) we have for sufficiently large ξ n . Now since σ satisfies (2.3) with α = 0, its x-derivatives are uniformly bounded, and hence for ǫ > 0 there exists an open neighbourhood V of the unit e of the group such that for all x · x −1 ξn ∈ V ⊆ G we have Let now u ∈ C ∞ (G) be such that u(x) = 0 for all x / ∈ V. Then u ξn (x) = 0 for all x ∈ x ξn V , i.e. for x · x −1 ξn / ∈ V. Then the last inequality following from (4.2). Therefore, using (4.5) and (4.6) for the last inequalities. Since σ(x ξn , ξ n )σ(x ξn , ξ n ) * is normal there exist a unitary matrix U such that with λ ii (x ξn , ξ n ) being the eigenvalues of σ(x ξn , ξ n )σ(x ξn , ξ n ) * . Now let We want to show that (4.8) u ξn (x)UΛU * 2 HS ≥ λ 2 mm u ξn (x) 2 HS . Let M = UΛU * . Then M is symmetric and λ mm (x ξn , ξ n ) is the minimum eigenvalue of the matrix M > 0. Let w ξn := u ξn (x)M. To prove (4.8) it is enough to show that w ξn HS ≥ λ mm (x ξn , ξ n ) w ξn M −1 HS , where M −1 = UΛ −1 U * . This is true since λ −1 mm (x ξn , ξ n ) is the maximum eigenvalue of M −1 , that is, M −1 op = λ −1 mm (x ξn , ξ n ). This proves (4.8).
Using (4.7) and (4.8), we can estimate Now, as ξ n → ∞, we have that is, for any ǫ > 0, Now, using the fact that ǫ is an arbitrary positive number, we have This completes the proof of Theorem 3.1.
Proof of Lemma 4.1. Let x, z ∈ G. Let us define

Now we can write
where z = y −1 x. Then from the definition of u ξn (x) = d −1/2 ξn ξ n (x)u(xx −1 ξn ) and (4.10) we obtain For a given collection of m strongly admissible difference operators ∆ 1 , . . . , ∆ m with the corresponding functions q 1 , . . . , q m ∈ C ∞ (G) with ∆ j f (ξ) = q j f (ξ) we have the Taylor expansion formula, see [RT10,RTW10], is the geodesic distance from x and e, and ∂ (α) x are some leftinvariant differential operators on G, and q α (x) = q 1 (x) α 1 · · · q m (x) αm . Assuming that u is sufficiently smooth, from the Taylor expansion formula we have Then by the left-invariance of ∂ Using (4.13) and (4.14), we can now write We denote and Then we have Calculating I 2 , we have And calculating I 3 , we have where we can denote q N := O(h(x) N ) so that q N vanishes at e of order N, but keep in mind that it is matrix-valued. Then we have and T 2 N := ∆ q N σ(x, ξ n ). We can estimate where z ∈ G. So, using (4.12) and (2.3), for some operator∂ where 1 ≤ |α| ≤ N, N is fixed and · H |α| is the Sobolev norm. Similarly, Therefore, as ξ n → ∞ we have T 1 N (x) HS → 0 and T 2 N (x) HS → 0 for all x ∈ G which gives HS dx → 0 as ξ n → ∞ and, similarly, T 2 N L 2 (G) → 0. This implies T σ u ξn − u ξn σ(·, ξ n ) L 2 (G) → 0 as ξ n → ∞, and we note that it is sufficient to take N = 1 in the above argument.

Remark 4.2.
Looking at what we have used in the proof, we note that we have (with the same proof and N = 1) the following extension of the Gohberg lemma without making an assumption that the operator belongs to Ψ 0 (G), namely: Let T σ : L 2 (G) → L 2 (G) be a bounded operator with the matrix symbol σ(x, ξ) satisfying, 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 is known as the Atkinson theorem which gives another equivalent definition of Fredholm operators.
Theorem 5.1. 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 we can find 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 .
We recall that the Wolf spectrum Σ w (A) of A is defined by Σ w (A) : Proof of Theorem 3.2. Let λ ∈ C be such that |λ| > d max . Then there exists ǫ > 0 such that |λ| > d max + ǫ. Now, by the definition of d max in Theorem 3.2, we have for some R > 0 and for all ξ ≥ R that sup Then for ξ ≥ R, we can estimate Hence from (2.4) it follows that the operator T σ − λI is elliptic and hence it is a Fredholm operator from L 2 (G) → L 2 (G), see e.g. [Hör07, Section 19.5]. Thus Therefore i(T σ − λI) = 0 for all {λ ∈ C : |λ| > d max }. This implies completing the proof of (3.3).
To prove the last part of Theorem 3.2, we start by recalling the definition of the Calkin algebra. Let L (L 2 (G)) and K (L 2 (G)) be respectively the C * algebra of bounded linear operators on L 2 (G) and the ideal of compact operators on L 2 (G). The Calkin algebra, L (L 2 (G))/K (L 2 (G)), is a * -algebra. The product and the involution are defined here as A − K L (L 2 (G)) , [A] ∈ L L 2 (G))/K (L 2 (G) .
By using the Calkin algebra the Gohberg lemma Theorem 3.1 can be reformulated as the inequality [T σ ] C ≥ d min .
We now prove the last part of Theorem 3.2. We assume that d max = 0 and observe that T σ is compact if and only if [T σ ] = 0 in the Calkin algebra L (L 2 (G))/K (L 2 (G)). We also observe that the operator T σ is essentially normal on L 2 (G), i.e. T σ T * σ −T * σ T σ is compact. Indeed, this is the operator of order −1 so the 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, where r(T σ ) is the spectral radius of [T σ ]. On the other hand we know that Σ ess (T σ ) ⊂ {0} by the first part of Theorem 3.2 in (3.3). This implies that T σ − λI is Fredholm for λ = 0. So using Atkinson's Theorem 5.1 this implies that there exists a bounded operator B such that (T σ − λI)B = I + K, where K is a compact operator. That is, for λ = 0, [(T σ − λI)] is invertible, which implies that for λ = 0, we have λ / ∈ Σ([T σ ]), the spectrum of [T σ ]. So Σ([T σ ]) ⊆ {0}. Consequently, [T σ ] C = r(T σ ) = 0. Therefore [T σ ] = 0, and hence T σ is compact, completing the proof.