QUANTITATIVE SPECTRAL THEORY OF COMPACT OPERATORS

TERENCE TAO

Abstract. The famous spectral theorem for compact operators T : X → X on a Banach space X asserts, among other things, that T has a discrete spectrum consisting entirely of generalized eigenspaces, with the only accumulation point being at zero. In this note we give a more quantitative formulation of this theorem, in which quantitative estimates on the resolvent for energies |λ| > r are given in terms of the r-entropy of the image of the unit ball under T.

1. Compactness

Let X be a complex Banach space with norm ∥∥_{X}, and let T : X → X be a bounded linear operator on X, thus the operator norm

∥T ∥_{X→X} := sup{∥T f∥_{X} : ∥f∥_{X} ≤ 1}

is finite. Let B(f, r) := {g ∈ X : ∥g − f∥_{X} < r} denote the open ball of radius r centered at f in X. Then clearly the image T B(0, 1) of the unit ball is bounded:

T B_{X}(0, 1) ⊆ B_{X}(0, ∥T ∥_{X→X}).

However, it does not necessarily ensure that T B(0, 1) is precompact (has compact closure). When the latter occurs, we say that T is a compact operator. Compact operators have a number of good properties, for instance they have a discrete spectrum of eigenvalues λ_{1}, λ_{2}, … tending to zero, and no other spectrum. In other words, we have Fredholm alternative: given any λ, either T − λ is invertible, or λ is an eigenvalue of T (i.e. there exists a non-zero f ∈ X such that T f = λf).

The purpose of this note is to explore more quantitative versions of these statements, in which we relate the amount of compactness of T to the number of eigenvalues, or of such quantities as ∥T − λ)^{−1}∥_{X→X}. The first thing to do is to understand how to quantify the concept of compactness. From metric space theory we know that a subset E of a metric space X is precompact if and only if it is totally bounded, i.e. for every r > 0, E can be covered by a finite number of balls B(x_{1}, r), …, B(x_{k}, r). An alternative definition is terms of entropies:

Definition 1.1. If E ⊂ X and r > 0, the r-entropy E_{r}(E) of E is the largest number of r-separated points one can find in E, or more precisely

E_{r}(E) := sup{N : there exist x_{1}, …, x_{N} ∈ E such that ∥x_{i}−x_{j}∥_{X} ≥ r for all 1 ≤ i < j ≤ N}.

Observe that a set E is totally bounded if and only if all its entropies are finite; thus an operator T is compact if and only if E_{r}(T B(0, 1)) < ∞ for all r > 0. Note also that the entropies are a non-increasing function of r, and that E_{r}(T B(0, 1)) will become finite if r is large enough (e.g. if r > 2∥T ∥_{X→X} then E_{r}(T B(0, 1)) = 1).

We now consider the question of what one can say about T if we do not assume that it is compact, but only that one of the entropies E_{r}(T B(0, 1)) is finite. Indeed, we shall set N := E_{r}(T B(0, 1)) and A := ∥T ∥_{X→X} and ask for what bounds one can then place on the spectrum and invertibility of T based on A, N and r. It turns out that under such assumptions, we have very little control on what the spectrum of T is doing in the ball {λ : |λ| ≤ r}, but we do have substantial control outside of this ball.

We turn to the details. For any r′ > r and δ > 0, we consider the following set Ω_{r′,δ} ⊂ X of “approximate eigenfunctions”:

Ω_{r′,δ} := {f ∈ X : ∥f∥_{X} = 1; ∥(T − λ)f∥_{X} ≤ δ for some |λ| ≥ r′}.

We can give a quantitative control for the “dimension” of the space of approximate eigenfunctions:

Lemma 1.2. Let 0 < ε ≤ \frac{1}{2}. If δ obeys the smallness condition

0 < δ < \frac{ε^{N+1}}{4(N + 1)}(r′ − r)  (1)

then Ω_{r′,δ} is contained in V_{r′,δ} + B(0, ε) for some finite-dimensional subspace V_{r′,δ} of dimension at most N.

Proof Fix ε, r′, δ. Suppose for contradiction that Ω_{r′,δ} were not contained in any set of the form V_{r′,δ} + B(0, ε). Then by the greedy algorithm one could find f_{0}, …, f_{N} ∈ Ω_{r′,δ} such that for each 0 ≤ n ≤ N, we have f_{n} ∉ V_{n} + B(0, ε), where V_{n} is the n-dimensional space spanned by f_{0}, …, f_{n−1}. This implies a quantitative linear independence of f_{0}, …, f_{N}:

Lemma 1.3. For any complex numbers c_{0}, …, c_{N}, we have

\frac{ε^{N+1}}{2(N + 1)} ∑_{n=0}^{N} |c_{n}| ≤ ∥c_{0}f_{0} + … + c_{N}f_{N}∥_{X} ≤ ∑_{n=0}^{N} |c_{n}|.

Proof The right-hand bound is just the triangle inequality, since ∥f_{n}∥_{X} = 1 for all n. Now we look at the left-hand side. We may normalize ∥c_{0}f_{0} + … + c_{N}f_{N}∥_{X} = 1. Since

c_{0}f_{0} + … + c_{N}f_{N} ∉ V_{N−1} + B(0, |c_{N}|ε)

we see that |c_{N}| ≤ 1/ε. Then ∥c_{0}f_{0}+…+c_{N−1}f_{N−1}∥_{X} ≤ 1+1/ε by the triangle inequality. Since

c_{0}f_{0} + … + c_{N−1}f_{N−1} ∉ V_{n−2} + B(0, |c_{N−1}|ε)

we obtain that |c_{N−1}| ≤ 1/ε + 1/ε^{2}. Continuing in this fashion we obtain

∑_{n=0}^{N} |c_{n}| ≤ ∑_{n=0}^{N} ∑_{m=1}^{N−m+1} ε^{−m} ≤ (N + 1) ∑_{m=1}^{N+1} ε^{m} ≤ 2(N + 1)ε^{N+1}

and the claim follows.

By definition of Ω_{r′,δ}, we can then find associated complex numbers λ_{0}, …, λ_{N} such that

∥f_{n}∥_{X} = 1; |λ_{n}| ≥ r′; T f_{n} ∈ λ_{n}f_{n} + B(0, δ).

Since V_{n} is finite-dimensional, we may find a g_{n} ∈ V_{n} for each n such that ∥f_{n} − g_{n}∥_{X} = ∥f_{n}∥_{X/V_{n}}. If we then set h_{n} := (f_{n} − g_{n})/∥f_{n} − g_{n}∥_{X}, then ∥h_{n}∥_{X} = ∥h_{n}∥_{X/V_{n}} = 1. Now, since h_{n} ∈ V_{n+1} we may write h_{n} = ∑_{m=0}^{n} c_{m,n}f_{m} for some scalars c_{m,n}; from Lemma 1.3 we have

∑_{m=0}^{n} |c_{m,n}| ≤ 2(N + 1)ε^{−N−1}.

Since T f_{n} ∈ λ_{n}f_{n} + B(0, δ), we thus have

T h_{n} ∈ ∑_{m=0}^{n} c_{m,n}λ_{m}f_{m} + B(0, 2(N + 1)ε^{−N−1}δ)

and so by comparing this to λ_{n}h_{n} we see that

T h_{n} ∈ λ_{n}h_{n} + V_{n} + B(0, 2(N + 1)ε^{−N−1}δ) ⊂ V_{n+1} + B(0, 2(N + 1)ε^{−N−1}δ).

In particular, for 0 ≤ m < n ≤ N we have

T h_{n} − T h_{m} ∈ λ_{n}h_{n} + V_{n} + B(0, 4(N + 1)ε^{−N−1}δ)

and hence

∥T h_{n} − T h_{m}∥_{X} ≥ ∥λ_{n}h_{n}∥_{X/V_{n}} − 4(N + 1)ε^{−N−1}δ ≥ r′ − 4(N + 1)ε^{−N−1}δ > r

by (1). But then h_{0}, …, h_{N} are a collection of N + 1 points in T B(0, 1) which are all r-separated, contradicting the definition of N.

Together with the space of approximate eigenfunctions, we can form a set of approximate eigenvalues:

Λ_{r′,δ} := {λ ∈ C : |λ| ≥ r′; ∥(T − λ)f∥_{X} ≤ δ for some ∥f∥_{X} = 1}.  (2)

The following lemma asserts in some sense that there are at most N approximate eigenvalues of magnitude larger than r.

Lemma 1.4. Let 0 < ε ≤ \frac{1}{2}. If δ obeys the smallness conditions

0 < δ ≤ min(A, \frac{1}{100(N + 1)}(\frac{ε}{100A})^{N^{2}+N}(r′ − r))  (3)

then E_{ε}(Λ_{r′,δ}) ≤ N.

Proof Suppose for contradiction that there exist λ_{0}, …, λ_{N} ∈ Λ_{r′,δ} such that |λ_{i}−λ_{j}| ≥ ε for all 0 ≤ i < j ≤ N. There thus exists functions f_{0}, …, f_{N} ∈ X with

∥f_{n}∥_{X} = 1; |λ_{n}| ≥ r′; T f_{n} ∈ λ_{n}f_{n} + B(0, δ).

In particular we have

|λ_{n}| = ∥λ_{n}f_{n}∥_{X} ≤ ∥T f_{n}∥_{X} + δ ≤ ∥T ∥_{X→X} + δ ≤ A + δ ≤ 2A.

In particular we may take r′ ≤ 2A since the claim is vacuous otherwise. Now set V_{n} := span(f_{0}, …, f_{n−1}). We claim that

∥f_{n}∥_{X/V_{n}} ≥ \frac{1}{2}(3A)^{−N}ε^{N}.  (4)

We prove this by induction. For n = 0 this is clear. Now suppose that n > 0 and the claim has already been proven for n−1. Since V_{n} is finite dimensional, there exists scalars c_{0}, …, c_{n−1} such that

∥f_{n}∥_{X/V_{n}} = ∥f_{n} − c_{0}f_{0} − … − c_{n−1}f_{n−1}∥_{X}.

4 TERENCE TAO

Since ∥f_{n}∥_{X/V_{n}} ≤ ∥f_{n}∥_{X} = 1, we see from the triangle inequality that

∥c_{0}f_{0} + … + c_{n−1}f_{n−1}∥_{X} ≤ 2

and hence by Lemma 1.3

∑_{m=0}^{n−1} |c_{m}| ≤ 4N(\frac{1}{2}(3A)^{−N}ε^{N})^{−N}.  (5)

Now consider the operator S := ∏_{j=0}^{n−1}(T − λ_{j}). This operator has norm

∥S∥_{X→X} ≤ ∏_{j=0}^{n−1} ∥T − λ_{j}∥_{X→X} ≤ ∏_{j=0}^{n−1}(A + |λ_{j}) ≤ (3A)^{N}

and so

∥f_{n}∥_{X/V_{n}} ≥ (3A)^{−N}∥Sf_{n} − c_{0}Sf_{0} − … − c_{n−1}Sf_{n−1}∥_{X}.

Since

T f_{n} ∈ λ_{n}f_{n} + B(0, δ)

we can show inductively that

∏_{j=0}^{m}(T − λ_{j})f_{n} ∈ ∏_{j=0}^{m}(λ_{n} − λ_{j})f_{n} + B(0, (6A)^{m}δ)

for all 0 ≤ m ≤ n, and thus

Sf_{n} ∈ ∏_{j=0}^{n−1}(λ_{n} − λ_{j})f_{n} + B(0, (6A)^{n−1}δ).

Similar considerations give

Sf_{m} ∈ B(0, (6A)^{n−1}δ).

Thus by (5)

Sf_{n} − c_{0}Sf_{0} − … − c_{n−1}Sf_{n−1} ∈ ∏_{j=0}^{n−1}(λ_{n} − λ_{j})f_{n} + B(0, 5N(\frac{1}{2}(3A)^{−N}ε^{N})^{−N}(6A)^{n−1}δ).

We thus have

∥f_{n}∥_{X/V_{n}} ≥ (3A)^{−N}(∏_{j=0}^{n−1} |λ_{n} − λ_{j}| − 5N(\frac{1}{2}(3A)^{−N}ε^{N})^{−N}(6A)^{N}δ)

≥ (3A)^{−N}(ε^{N} − 5(\frac{1}{2}(3A)^{−N}ε^{N})^{−N}(6A)^{n−1}δ)

≥ \frac{1}{2}(3A)^{−N}ε^{N}

by (1). This closes the induction. But now we can repeat the proof of (1.2) (with \frac{1}{2}(3A)^{−N}ε^{N} playing the role of ε) to obtain a contradiction as before.

This lemma already gives us good control on the spectrum

σ(T) := {λ ∈ C : T − λ is not boundedly invertible}.

Lemma 1.5. The set {λ ∈ σ(T) : |λ| > r} is finite, with cardinality at most N.

Proof It suffices to show that for every r′ > r, the set E = {λ ∈ σ(T) : |λ| ≥ r′} has cardinality at most N. As is well known, the boundedness of T implies that the spectrum σ(A), and hence A is compact. We will show that the boundary ∂E of E has cardinality at most N, which forces E itself to have cardinality at most N. If λ_{0} ∈ ∂E, then there exists a sequence λ_{n} converging to λ_{0} with |λ_{n}| ≥ r and T −λ_{n} invertible. This forces ∥(T − λ_{n})^{−1}∥_{X→X} to go to infinity, otherwise we could invert ∥(T − λ_{0})^{−1}∥_{X→X} by Neumann series. Thus there exist functions f_{n} with

∥f_{n}∥_{X} = 1; ∥(T − λ_{n})^{−1}f_{n}∥_{X} → 0.

Comparing this with the definition of Λ_{r′,δ}, we see that λ_{0} lies in the closure of Λ_{r′,δ} for every δ > 0. By Lemma 1.4 this means that E_{ε}(∂E) ≤ N for all ε > 0, which implies that ∂E has cardinality at most N, as desired.

A related fact is:

Lemma 1.6 (Fredholm alternative). If λ ∈ σ(T) and |λ| > r, then λ is an eigenvalue of T, i.e. there exists a non-zero f ∈ X such that T f = λf.

Proof We recall the sequence λ_{n} and f_{n} defined in the previous lemma. Observe from Lemma 1.2 that for any ε > 0, the sequence f_{n} will eventually be contained in the εneighbourhood of a finite-dimensional space. Since the sequence f_{n} is also bounded, we thus see that it is totally bounded, hence precompact. If we then extract a convergent subsequence of f_{n} converging to some f and take limits we obtain T f − λf = 0 and ∥f∥_{X} = 0, and the claim follows.

Now suppose that λ is not in the spectrum. Then we know abstractly that T − λ is boundedly invertible, but the above argument does not directly give any quantitative estimates on the inverse. However, this can be accomplished by appealing again to Lemma 1.4 and using some complex analysis:

Lemma 1.7. Let Σ := σ(T) ∪ {λ : |λ| ≤ r}. Then for every λ ∉ Σ with |λ| ≤ 2A, we have

∥(T − λ)^{−1}∥_{X→X} ≤ \frac{1}{A}(\frac{400A(N + 1)}{dist(λ, Σ)})^{N^{2}+N+1}.

Proof Set d := dist(λ, Σ)/2 and

δ := A(\frac{d}{200A(N + 1)})^{N^{2}+N+1}.

It will suffice to show that

|R_{f,g}(λ)| ≤ 1/δ

for all f ∈ X, g ∈ X^{∗} with ∥f∥_{X} = 1, ∥g∥_{X^{∗}} = 1, where R_{f,g} is the resolvent function

R_{f,g}(λ) := ⟨(T − λ)^{−1}f, g⟩

defined on the complement of the spectrum σ(T).

Suppose for contradiction that |R_{f,g}(λ)| > 1/δ. Observe that R_{f,g} is holomorphic on the complement of σ(T), and in particular on the disk of radius d centered at λ. Then by the maximum principle, we know that for every radius 0 < d′ < d there exists λ′ with |λ′ − λ| = d′ such that |R_{f,g}(λ′)| > 1/δ. In particular, this implies

E_{d/2(N+1)}({λ′ : |λ′ − λ| ≤ d; |R_{f,g}(λ′)| ≥ 1/δ}) ≥ N + 1.

On the other hand, if |R_{f,g}(λ′)| ≥ 1/δ then ∥(T − λ′)^{−1}∥_{X→X} ≥ 1/δ and |λ′| ≥ r + d, which by (2) implies that λ ∈ Λ_{r+d,δ}. But this contradicts Lemma 1.4 by choice of δ.

For each λ, let E(λ) := {f ∈ X : (T − λ)^{n}f = 0 for some n ≥ 0} be the generalized eigenspace of λ. We already know that there are at most N eigenvalues with magnitude greater than r; now we prove the same statement, counting multiplicity.

Lemma 1.8. If we let V := ∑_{λ∈σ(T):|λ|>r} E(λ), then V is a finite-dimensional space with dimension at most N.

Proof Suppose for contradiction that V had dimension at least N + 1 By counting eigenvectors (i.e. zeroes of (T −λ)) first, and then second-order eigenvectors (zeroes of (T − λ)^{2}), and so forth, one can then find a sequence f_{0}, …, f_{N} of linearly independent vectors in V and eigenvalues λ_{0}, …, λ_{N} such that for each n, (T − λ_{n})f_{n} ∈ V_{n}, where V_{n} as usual is the n-dimensional space spanned by f_{0}, …, f_{n−1}. By repeating the proof of Lemma 1.2 (and noting that δ is essentially zero here) we can eventually show that {f_{0}, …, f_{n}} is contaqined in the ε-neighbourhood of an N-dimensional vector space for any ε, which leads to a contradiction for ε sufficiently small thanks to the linear independence of the f_{n}.

Let us take a closer look at the finite-dimensional space V in the above lemma. Since V is the direct sum of generalized eigenspaces, we know that V is an invariant subspace of T (i.e. T V ⊆ V). In particular, T : X → X descends to an operator T̃ : X/V → X/V on the quotient space X/V, defined in the usual manner as the space of cosets f + V of V endowed with the norm

∥f + V ∥_{X/V} := inf{∥f + g∥_{X} : g ∈ V },

and T̃(x + V) := T x + V (this is well-defined since V is invariant). It is easy to verify the inequality

E_{r}(T̃B_{X/V} (0, 1)) ≤ E_{r}(T B_{X}(0, 1)) = N.

Also, the operator T̃ can have no eigenvalues of magnitude greater than r, since otherwise they would have been placed in V. Thus by Lemma 1.6, T̃ has no spectrum in the exterior region {λ ∈ C : |λ| > r}. By Lemma 1.7, we then have the estimate

∥(T̃ − λ)^{−1}∥_{X/V →X/V} ≤ \frac{1}{A}(\frac{400A(N + 1)}{|λ| − r})^{N^{2}+N+1} whenever r < |λ| ≤ 2A.  (6)

To relate this back to T, we use the following general lemma:

Lemma 1.9. Let suppose that S : X → X is a bounded linear operator with invariant subspace V such that the induced mapping S̃ : X/V → X/V is boundedly invertible. Then S is boundedly invertible on X if and only if it is boundedly invertible on V, and in that case we have the estimates

∥S^{−1}∥_{V →V} ≤ ∥S^{−1}∥_{X→X} ≤ ∥S^{−1}∥_{V →V} +∥S^{−1}∥_{V →V} ∥S∥_{X→X}∥S̃^{−1}∥_{X/V →X/V} +∥S̃^{−1}∥_{X/V →X/V} .

Proof It is clear that if S is boundedly invertible on X then it must preserve V (otherwise the map S : X/V → X/V would have a nontrivial kernel) and hence will be boundedly invertible on V; since S^{−1} : V → V is simply a restriction of S^{−1} : X → X we have ∥S^{−1}∥_{V →V} ≤ ∥S^{−1}∥_{X→X}.

Now suppose that S is boundedly invertible on V. Then S : V → V and S̃ : X/V → X/V are both bijective, which implies that S : X → X is also bijective. Now let f ∈ X and ε > 0. Then

∥S^{−1}f + V ∥_{X/V} ≤ ∥S̃^{−1}∥_{X/V →X/V} ∥f + V ∥_{X/V} ≤ ∥S̃^{−1}∥_{X/V →X/V} ∥f∥_{X}

so there exists h ∈ V such that

∥S^{−1}f − h∥_{X} ≤ ∥S̃^{−1}∥_{X/V →X/V} ∥f∥_{X} + ε.  (7)

Applying S, we obtain

∥f − Sh∥_{X} ≤ ∥S∥_{X→X}∥S̃^{−1}∥_{X/V →X/V} ∥f∥_{X} + ∥S∥_{X→X}ε.

and hence by the triangle inequality

∥Sh∥_{X} ≤ (1 + ∥S∥_{X→X}∥S̃^{−1}∥_{X/V →X/V})∥f∥_{X} + ∥S∥_{X→X}ε.

Applying S^{−1} : V → V, and then combining with (7) by the triangle inequality again, we obtain

∥S^{−1}f∥_{X} ≤ (∥S^{−1}∥_{V →V} +∥S^{−1}∥_{V →V} ∥S∥_{X→X}∥S̃^{−1}∥_{X/V →X/V} +∥S̃^{−1}∥_{X/V →X/V})∥f∥_{X}+∥S^{−1}∥_{V →V} ∥S∥_{X→X}ε.

Since ε was arbitrary, the claim follows.

We thus conclude

∥(T − λ)^{−1}∥_{X→X} ≤ \frac{1}{A}(\frac{400A(N + 1)}{|λ| − r})^{N^{2}+N+1}(1 + 4A∥(T − λ)^{−1}∥_{V →V})

for all r < |λ| ≤ 2A. (For |λ| > 2A one has the better bound ∥(T − λ)^{−1}∥_{X→X} ≤ 1/(|λ| − A) coming from Neumann series). This estimate clearly shows how the singularities of the resolvent (T − λ)^{−1} in the exterior disk |λ| > r are completely controlled by the behavior of this resolvent on the finite dimensional space V, which is of course essentially just a rational matrix-valued function of degree at most N; this is also a quantitative realization of the Fredholm alternative, which asserts (given sufficient compactness) that the invertibility of T − λ is equivalent to the absence of eigenfunctions. The estimate

(6) reinforces this picture by showing that once V is quotiented out, no obstructions to invertibility remain; this in particular is a quantitative version of the statement that the spectrum for energies |λ| > r is completely contained in V.

As a corollary we also can obtain the usual spectral theorem for compact operators T : X → X, since for such operators E_{r}(T B(0, 1)) is finite for every r > 0.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

  E-mail address: tao@math.ucla.edu