Pseudospectra of operator polynomials

Steffen Roch

Abstract

The recent interest in ε-pseudospectra of operators results from their (in comparison with usual spectra) excellent continuity properties. The goal of the present paper is to introduce and to examine ε-pseudospectra of operator polynomials with main emphasis on the continuity aspect.

1 Introduction

Let H be a separable Hilbert space, L(H) the C^{∗}-algebra of all linear and bounded operators on H, and A ∈ L(H) an operator on H with matrix representation (a_{ij})_{i,j=1}^{∞} with respect to some fixed basis of H. The nth finite section (a_{ij})_{i,j=1}^{n} of A will be denoted by A_{n}. What can be said about the relation between the spectrum σ(A) of the operator A and the eigenvalues of the matrix A_{n} as n tends to infinity?

In case the operator A is self-adjoint and has a connected spectrum, the answer is affirmative: In this case the spectra of A_{n} converge to the spectrum of A in the sense that

lim σ_{ℂ^{n×n}}(A_{n}) = σ_{L(H)}(A)

where here and hereafter the following notation is used: Given a sequence (M_{n}) of subsets of the complex plane, let lim M_{n} denote the set of all partial limits of sequences (m_{n}) of points m_{n} ∈ M_{n}. On the other hand, this convergence can fail drastically in case the operator A is not normal. To have at least one (trivial) example, consider the shift operator A = (a_{ij})_{i,j=1}^{∞} where a_{ij} = 1 if i − j = 1 and a_{ij} = 0 otherwise. The spectrum of A is the closed unit disk {z ∈ ℂ : |z| ≤ 1}, whereas the spectrum of every of its finite sections A_{n} consists of the point 0 only.

Reichel and Trefethen [8, 12, 13] observed that, for several classes of operators, the convergence behaviour becomes much better if the usual spectra are replaced by a certain approximate spectrum which was introduced by Landau [6, 7], and which is known as the ε-pseudospectrum. By definition, given ε > 0, the εpseudospectrum σ^{(ε)}(A) of the operator A ∈ L(H) consists of all λ ∈ ℂ such that either A − λI is not invertible, or it is invertible, but ∥(A − λI)^{−1}∥ ≥ 1/ε.

Analogously one defines the ε-pseudospectrum of an element of a C^{∗}-algebra with identity.

A computer working with finite accuracy cannot distinguish between a noninvertible matrix and an invertible matrix the inverse of which has a very large norm. Thus, the definition of pseudospectra reflects in some sense that finite accuracy.

The usual spectrum follows from its ε-counterpart by letting ε formally go to zero. More precise:

σ(A) = ∩_{ε>0}σ^{(ε)}(A).

As usual spectra, ε-pseudospectra are compact and non-empty subsets of the complex plane. Also the upper semi-continuity of spectra has its analogue for pseudospectra: If (A_{n}) ⊆ L(H) is a norm-convergent sequence of operators, then

lim σ^{(ε)}(A_{n}) ⊆ σ^{(ε)}(lim A_{n}),

which can be easily checked.

The results of Reichel and Trefethen, which had been extended later by Böttcher [1], concern pseudospectra of Toeplitz operators and of their finite sections. Given a function a ∈ L^{∞}(𝕋) with Fourier coefficients a_{k}, k ∈ ℤ, the Toeplitz operator T (a) generated by a is given by the matrix (a_{i−j})_{i,j=1}^{∞} thought of as acting on the Hilbert space l^{2} provided with its standard basis. In case the function a belongs to the Wiener algebra, i.e. if ∑ |a_{k}| < ∞, Reichel and Trefethen verified that

lim_{n→∞} σ^{(ε)}(T_{n}(a)) = σ^{(ε)}(T (a))  (1)

for every ε > 0. Böttcher extended this result to the case of arbitrary piecewise continuous functions a. A quite general approach to the study of the asymptotic behaviour of pseudospectra, which bases essentially on Böttcher’s ideas, was proposed in [9].

The goal of the present paper is to extend results such as (1) to the case of operator polynomials. Thereby, the general approach of [9] will serve as a guide.

2 Pseudospectra of operator polynomials

The following considerations are not restricted to operators on a Hilbert space; so let B be an arbitrary C^{∗}-algebra with identity e. Given elements b_{0}, b_{1}, …, b_{m} of B, the expression

L(λ) := b_{0} + b_{1}λ + … + b_{m}λ^{m}

is well-defined for every complex number λ, and L is (not quite correctly) called an operator polynomial with coefficients in B. The spectrum of L is the set σ(L) of all λ ∈ ℂ for which L(λ) is not invertible; similarly, for ε > 0, the ε-pseudospectrum of L is the set σ^{(ε)}(L) of all λ ∈ ℂ such that either L(λ) is not invertible, or ∥L(λ)^{−1}∥ ≥ 1/ε. Evidently, if L(λ) = b − λe, then the spectrum and the εpseudospectrum of the polynomial L coincide with the common spectrum and ε-pseudospectrum of the element b ∈ B, respectively.

The following result establishes another characterization of pseudospectra of operator polynomials. Its proof follows that for the special case L(λ) = b − λe which belongs to Ehrhardt and Finck and is published in [9].

Theorem 1 Let B be a unital C^{∗}-algebra and ε > 0. Then, for every operator polynomial L with coefficients in B,

σ^{(ε)}(L) = {λ ∈ ℂ : there is a p ∈ B with ∥p∥ ≤ ε and t ∈ σ(L + p)}. (2)

Clearly, L + p refers to the polynomial (L + p)(λ) = (b_{0} + p) + b_{1}λ + … + b_{m}λ^{m}.

Proof. Abbreviate the sets on the left and right hand side of (2) by S_{1} and S_{2}, respectively. The first claim is the inclusion S_{2} ⊆ S_{1}.

Given t ∈ S_{2}, choose p in B with ∥p∥ ≤ ε such that t ∈ σ(L + p). If t already lies in σ(L), then it also belongs to S_{1}, and we are done. So suppose t ∉ σ(L). Then L(t) is invertible, and the identity

L(t) + p = L(t)(e + L(t)^{−1}p)

yields that e + L(t)^{−1}p cannot be invertible. Hence, ∥L(t)^{−1}p∥ ≥ 1 (otherwise invertibility would follow via Neumann series). Because of

1 ≤ ∥L(t)^{−1}p∥ ≤ ∥L(t)^{−1}∥ ∥p∥

one has

∥L(t)^{−1}∥ ≥ 1/∥p∥ ≥ 1/ε,

i.e. t ∈ S_{1} in this case, too. For the reverse inclusion S_{1} ⊆ S_{2}, assume there is a t ∈ S_{1} such that L(t) + p is invertible for all p ∈ B with ∥p∥ ≤ ε. Choosing p = 0 one particularly gets the invertibility of L(t) and, hence, of L(t)^{∗} := b_{0}^{∗} + b_{1}^{∗}t̄+ … + b_{n}^{∗}t̄^{n}, and, choosing p = λ(L(t)^{∗})^{−1} with a certain complex number λ which satisfies

0 < |λ| ≤ ε/∥(L(t)^{∗})^{−1}∥,  (3)

one obtains invertibility of

L(t) + λ(L(t)^{∗})^{−1} = λL(t) (\frac{1}{λ}e + L(t)^{−1}(L(t)^{∗})^{−1})

(observe that ∥p∥ ≤ ε in both cases). Hence, \frac{1}{λ}e + L(t)^{−1}(L(t)^{∗})^{−1} is invertible for all λ satisfying (3). This gives

ρ(L(t)^{−1}(L(t)^{∗})^{−1}) < ∥(L(t)^{∗})^{−1}∥ /ε,

whence via the self-adjointness of L(t)^{−1}(L(t)^{∗})^{−1} (implying the coincidence of spectral radius ρ and norm) follows

∥L(t)^{−1}∥^{2} = ∥L(t)^{−1}(L(t)^{∗})^{−1}∥ < ∥(L(t)^{∗})^{−1}∥ /ε.

Finally one has ∥L(t)^{−1}∥ = ∥(L(t)^{∗})^{−1}∥ which yields ∥L(t)^{−1}∥ < 1/ε in contrast to t ∈ S_{1}. This contradiction verifies the inclusion S_{1} ⊆ S_{2}.

In case B = ℂ^{n×n}, Theorem 1 has been widely used for numerical computations of ε-pseudospectra by randomly choosing n×n matrices p with ∥p∥ ≤ ε and then plotting the eigenvalues of a + p for a ∈ ℂ^{n×n}. Beautiful plots of pseudospectra of Toeplitz matrices can be found in [1, 3, 8, 12, 13].

Observe furthermore that if B is a unital C^{∗}-algebra and A a C^{∗}-subalgebra of B which contains the identity, then

σ_{A}^{(ε)}(L) = σ_{B}^{(ε)}(L) for every polynomial L with coefficients in A,

i.e. C^{∗}-algebras are inverse closed with respect to ε-pseudo-invertibility.

3 Limiting sets of pseudospectra

The goal of this section is to derive a description of the limiting set lim σ^{(ε)}(L_{n}) of the pseudospectra of a given sequence of operator polynomials L_{n}. Again, it is not essential that the coefficients of the L_{n} are operators; so assume that C_{1}, C_{2}, … is a sequence of unital C^{∗}-algebras, and let F stand for the set of all bounded sequences (c_{n}) where c_{n} ∈ C_{n}. Further let m > 0, and let (a_{n}^{(0)}), (a_{n}^{(1)}), …, (a_{n}^{(m)}) be given sequences in F. For every n, consider the polynomial with coefficients in C_{n},

L_{n}(λ) := a_{n}^{(0)} + a_{n}^{(1)}λ + … + a_{n}^{(m)}λ^{m}.  (4)

The desired description will consist in identifying the limiting set lim σ^{(ε)}(L_{n}) with the ε-pseudospectrum of a certain element of an accordingly chosen C^{∗}- algebra. A natural choice for that algebra (which also plays a prominent role for other problems in numerical analysis) is as follows. First observe that the set F forms a C^{∗}-algebra itself when this set is provided with elementwise operations, an elementwise involution, and the supremum norm. Moreover, the set of all sequences (c_{n}) ∈ F with ∥c_{n}∥ → 0 as n → ∞ is a closed ideal of F which will be denoted by G. It will be pointed out that the quotient algebra F/G is the needed one. Thus, it should be possible to identify the limiting set of the pseudospectra of the polynomials L_{n} with the pseudospectrum of something in F/G. It turns out that one can indeed do this, and that the something is the polynomial

L(λ) := ((a_{n}^{(0)}) + G) + ((a_{n}^{(1)}) + G)λ + … + ((a_{n}^{(m)}) + G)λ^{m}  (5)

with coefficients in F/G. The precise statement is in the following theorem. Recall that a sequence (c_{n}) ∈ F is said to be stable if there is an n_{0} such that all elements c_{n} with n ≥ n_{0} are invertible and sup_{n≥n_{0}} ∥c_{n}^{−1}∥ < ∞ or, equivalently, if the coset (c_{n}) + G is invertible in the quotient algebra F/G.

Theorem 2 Let m > 0, let (a_{n}^{(0)}), (a_{n}^{(1)}), …, (a_{n}^{(m)}) be sequences in F with (a_{n}^{(m)}) being stable, and let L_{n} and L be the polynomials (4) and (5). Then, for every ε > 0,

lim σ_{C_{n}}^{(ε)}(L_{n}) = σ_{F/G}^{(ε)}(L).

The proof is based on the following generalization of an observation by Daniluk (see [1] or [9]), concerning the special case L(λ) = b_{0} − λe, which is of its own interest.

Theorem 3 Let B be a C^{∗}-algebra with identity e, let m > 0, and let L be the polynomial L(λ) = b_{0}+b_{1}λ+…+b_{m}λ^{m} with coefficients in B where the coefficient b_{m} is supposed to be invertible. Assume further there is an open subset U of the complex plane such that L(λ) is invertible and ∥L(λ)^{−1}∥ ≤ M for all λ ∈ U. Then ∥L(λ)^{−1}∥ < M for all λ ∈ U.

In other words: the analytic function U → B, λ ↦ L(λ)^{−1} satisfies the maximum principle. This is a surprising result since – in contrast to complex-valued analytic functions – the maximum principle for operator-valued analytic functions does not hold in general as already the simple example

\{z ∈ ℂ : |z| < 1\} → ℂ^{2×2}, λ ↦ ( \begin{matrix}λ & 0 \\ 0 & 1 \end{matrix} )

indicates. Proof of Theorem 3. The proof will be subdivided into several steps; the third one being identical with Daniluk’s original proof. Step 1. Suppose there exists a λ_{0} ∈ U such that L(λ_{0}) = M. For the shifted polynomial

Q(λ) := L(λ + λ_{0}) = a_{0} + a_{1}λ + … + a_{m}λ^{m},

one easily checks that Q(λ) is invertible and ∥Q(λ)^{−1}∥ ≤ M for all λ ∈ (U − λ_{0}) (= the algebraic difference), and that ∥Q(0)^{−1}∥ = M. Moreover, both a_{m} = b_{m} and a_{0} = Q(0) are invertible. Further, let Q̂ refer to the monic polynomial defined by Q(λ) = Q̂(λ)a_{m}. Clearly, Q̂(λ) is invertible if and only if Q(λ) is so. Step 2. Here some facts concerning a special representation called linearization of the inverse of a monic operator polynomial are recalled. For details see [10], Chapter 2, Theorem 2.5.2.

For the desired representation, introduce the following vectors of length m and matrices of order m × m with entries in the algebra B:

X_{0} := ( e 0 0 ⋯ 0 ) , I := diag ( e e e ⋯ e) ,

Y := ( 0 ; 0 ; ⋮ ; 0 ; e ) , T := ( \begin{matrix}0 & e & 0 & ⋯ & 0 \\ 0 & 0 & e & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ −a_{0}a_{m}^{−1} & −a_{1}a_{m}^{−1} & −a_{2}a_{m}^{−1} & ⋯ & −a_{m−1}a_{m}^{−1} \end{matrix} ) ,

i.e. T is the companion matrix of the monic polynomial Q̂. Then, for all λ which are not in the spectrum of Q̂,

Q̂(λ)^{−1} = X_{0}(λI − T)^{−1}Y.

Since Q(λ)^{−1} = a_{m}^{−1}Q̂(λ)^{−1}, this yields the representation

Q(λ)^{−1} = X(λI − T)^{−1}Y  (6)

with X = a_{m}^{−1}X_{0} = ( a_{m}^{−1} 0 0 ⋯ 0 ), which holds for all λ such that Q(λ) is invertible. Step 3. Beginning with this step, it will be advantegeous to think of B as an algebra of linear bounded operators acting on some Hilbert space H with scalar product ⟨., .⟩ which is possible due to the GNS construction. Thus, T is actually an operator acting on the orthogonal sum of m exemplaries of H.

Since Q(0) is invertible, the operator T is invertible; hence, for all λ in the disk |λ| ≤ r with sufficiently small radius r,

(T − λI)^{−1} = ∑_{j=0}^{∞} λ^{j}T^{−j−1},

whence

X(T − λI)^{−1}Y = ∑_{j=0}^{∞} λ^{j}XT^{−j−1}Y

follows. Thus, for all f ∈ H and |λ| ≤ r,

∥Q(λ)^{−1}f∥^{2} = ∥X(T − λI)^{−1}Y f∥^{2} = ∑_{j,k≥0} λ^{j}\overline{λ}^{k}⟨XT^{−j−1}Y f, XT^{−k−1}Y f⟩.

Integrating this identity with respect to λ against the circle |λ| = r yields

\frac{1}{2πr} ∫_{|λ|=r} ∥Q(λ)^{−1}f∥^{2} |dλ| = ∑_{j≥0} r^{2j}∥XT^{−j−1}Y f∥^{2}.

Because of ∥Q(λ)^{−1}f∥ ≤ M∥f∥ by hypothesis, this yields for all j ≥ 1 the estimate

∥XT^{−1}Y f∥^{2} + r^{2j}∥XT^{−j−1}Y f∥^{2} ≤ ∑_{j≥0} r^{2j}∥XT^{−j−1}Y f∥^{2} ≤ M^{2}∥f∥^{2}. (7)

Let ε > 0 be arbitrarily given. Since, by assumption, ∥Q(0)^{−1}∥ = ∥XT^{−1}Y ∥ = M, there is an f_{ε} in H with norm 1 such that

∥XT^{−1}Y f_{ε}∥^{2} ≥ M^{2} − ε^{2},

which together with estimate (7) shows that M^{2} − ε^{2} + r^{2j}∥XT^{−j−1}Y f_{ε}∥^{2} ≤ M^{2} or

∥XT^{−j−1}Y f_{ε}∥ ≤ εr^{−j}  for all j ≥ 1.  (8)

Step 4. Set for brevity c_{j} = −a_{i}a_{m}^{−1}. The operator c_{0} is invertible, and it is easy to check that the inverse of the companion matrix T is a companion matrix again:

T^{−1} = ( \begin{matrix}−c_{0}^{−1}c_{1} & −c_{0}^{−1}c_{2} & ⋯ & −c_{0}^{−1}c_{n−1} & c_{0}^{−1} \\ e & 0 & ⋯ & 0 & 0 \\ 0 & e & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & e & 0 \end{matrix} ) .

Computing step by step the last columns of the operator matrices T^{−2},

T^{−2} + c_{0}^{−1}c_{1}T^{−1},
T^{−1}(T^{−2} + c_{0}^{−1}c_{1}T^{−1}) = T^{−3} + c_{0}^{−1}c_{1}T^{−2},
T^{−3} + c_{0}^{−1}c_{1}T^{−2} + c_{0}^{−1}c_{2}T^{−1},

one obtains

( −c_{0}^{−1}c_{1}c_{0}^{−1} ; c_{0}^{−1} ; 0 ; ⋮ ; 0 ) , ( 0 ; c_{0}^{−1} ; 0 ; ⋮ ; 0 ) , ( −c_{0}^{−1}c_{2}c_{0}^{−1} ; 0 ; c_{0}^{−1} ; ⋮ ; 0 ) , ( 0 ; 0 ; c_{0}^{−1} ; ⋮ ; 0 ) ,

respectively, what finally shows that the last columns of the matrices T^{−m} + c_{0}^{−1}c_{1}T^{−m+1} + … + c_{0}^{−1}c_{m−1}T^{−1},

T^{−m−1} + c_{0}^{−1}c_{1}T^{−m} + … + c_{0}^{−1}c_{m−1}T^{−2}

are as follows:

( 0 ; 0 ; ⋮ ; 0 ; c_{0}^{−1} ) , ( c_{0}^{−2} ; 0 ; ⋮ ; 0 ; 0 ) .

Consequently,

X(T^{−m−1} + c_{0}^{−1}c_{1}T^{−m} + … + c_{0}^{−1}c_{m−1}T^{−2})Y = a_{m}^{−1}c_{0}^{−2} = a_{0}^{−1}a_{m}a_{0}^{−1}. (9)

Step 5. Identity (9) implies that

∥a_{0}^{−1}a_{m}a_{0}^{−1}f_{ε}∥ = ∥X(T^{−m−1} + c_{0}^{−1}c_{1}T^{−m} + … + c_{0}^{−1}c_{m−1}T^{−2})Y f_{ε}∥,

and since Xc_{0}^{−1}c_{j} = a_{m}^{−1}c_{0}^{−1}c_{j}a_{m}X for all j, one further concludes that

∥a_{0}^{−1}a_{m}a_{0}^{−1}f_{ε}∥

= ∥(XT^{−m−1}Y + a_{m}^{−1}c_{0}^{−1}c_{1}a_{m}XT^{−m}Y + … + a_{m}^{−1}c_{0}^{−1}c_{m−1}a_{m}XT^{−2}Y)f_{ε}∥

≤ ∥XT^{−m−1}Y f_{ε}∥ + ∥a_{m}^{−1}c_{0}^{−1}c_{1}∥ · ∥XT^{−m}Y f_{ε}∥ + … +
+ ∥a_{m}^{−1}c_{0}^{−1}c_{m−1}a_{m}∥ · ∥XT^{−2}Y f_{ε}∥.

Together with (8) this yields

∥a_{0}^{−1}a_{m}a_{0}^{−1}f_{ε}∥ ≤ (\frac{1}{r^{m}} + \frac{a_{m}^{−1}c_{0}^{−1}c_{1}a_{m}}{r^{m−1}} + ⋯ + \frac{a_{m}^{−1}c_{0}^{−1}c_{m−1}a_{m}}{r} ) ε =: Cε

with a constant C depending on the polynomial Q and the radius r only. Letting ε go to zero in

1 = ∥f_{ε}∥ ≤ ∥a_{0}a_{m}^{−1}a_{0}∥ ∥a_{0}^{−1}a_{m}a_{0}^{−1}f_{ε}∥ ≤ C∥a_{0}a_{m}^{−1}a_{0}∥ ε

one arrives at a contradiction.

Having this result at disposal, one can prove Theorem 2 essentially by repeating arguments from Böttcher’s paper [1] (although he did not mention the corresponding special case of Theorem 2).

Proof of Theorem 2. Let λ ∈ σ_{F/G}^{(ε)}(L). Then either L(λ) is not invertible in F/G, or the inverse of this coset exists, but ∥L(λ)^{−1}∥ ≥ 1/ε.

In the first case, the sequence

(a_{n}^{(0)} − a_{n}^{(1)}λ + … + a_{n}^{(m)}λ^{m})_{n≥1} = (L_{n}(λ))_{n≥1}

fails to be stable whence easily follows that there is an infinite subsequence (n_{k}) of ℕ such that λ ∈ σ_{C_{n_{k}}}^{(ε)} (L_{n_{k}}) for every k. Clearly, λ ∈ lim σ^{(ε)}(L_{n}) in this case.

In the second case, Theorem 3 implies that, in every open neighbourhood U of λ, there is an λ_{0} such that ∥L(λ_{0})^{−1}∥ > 1/ε (otherwise one would have ∥L(λ_{0})−1∥ ≤ 1/ε for all λ_{0} ∈ U which would imply via Theorem 3 that ∥L(λ_{0})^{−1}∥ is strictly less than 1/ε for all λ_{0} ∈ U including λ_{0} = λ.)

Thus, for all sufficiently large k, there are numbers λ_{k} with λ_{k} → λ as k → ∞ such that

∥L(λ_{k})^{−1}∥ ≥ 1/(ε − 1/k).

It is an easy exercise (see also [1]) to check that this is equivalent to

lim sup_{n→∞} ∥L_{n}(λ_{k})^{−1}∥ ≥ 1/(ε − 1/k)

(with the invertibility of L_{n}(λ_{k}) for all sufficiently large n being a consequence of a Neumann series argument). Since 1/ε < 1/(ε − 1/k), there are numbers n_{k} tending to infinity as k → ∞ such that

∥L_{n_{k}}(λ_{k})^{−1}∥ ≥ 1/ε for all k;

in other words, λ_{k} ∈ σ^{(ε)}(L_{n_{k}}) for every sufficiently large k, which implies that λ = lim λ_{k} belongs lim σ^{(ε)}(L_{n}).

For the reverse inclusion, let λ ∈ lim σ^{(ε)}(L_{n}), but assume for contrary that λ ∉ σ_{F/G}^{(ε)}(L). Thus, L(λ) is invertible in F/G and ∥L(λ)^{−1}∥ = 1/ε−2δ < 1/ε with some δ > 0. As already mentioned, then the L_{n}(λ) are invertible for sufficiently large n and, by the lim sup formula for the norm in F/G again,

lim sup_{n→∞} ∥L_{n}(λ)^{−1}∥ = 1/ε − 2δ

whence follows that ∥L_{n}(λ)^{−1}∥ < 1/ε − δ for n sufficiently large, say n ≥ n_{0}.

The uniform boundedness of the coefficient sequences (a_{n}^{(1)}), (a_{n}^{(2)}), … (a_{n}^{(m)}) implies the existence of a constant C independent of n such that

∥L_{n}(λ) − L_{n}(μ)∥ ≤ C|λ − μ| for all μ ∈ U_{1}(λ).

Hence, there exists an open neighbourhood U ⊆ U_{1}(λ) which is independent of n such that

∥L_{n}(λ) − L_{n}(μ)∥ ≤ C|λ − μ| ≤ \frac{εδ}{1/ε − δ} for all μ ∈ U.

A Neumann series argument shows that then, for all μ ∈ U,

∥L_{n}(μ)^{−1}∥ = ∥L_{n}(λ)^{−1}(e − (L_{n}(λ) − L_{n}(μ))L_{n}(λ)^{−1})^{−1}∥
≤ \frac{∥L_{n}(λ)^{−1}∥}{1 − ∥L_{n}(λ) − L_{n}(μ)∥ ∥L_{n}(λ)^{−1}∥}
< \frac{1/ε − δ}{1 − εδ(1/ε − δ)^{−1}(1/ε − δ)} = \frac{1}{ε},

hence μ ∉ σ^{(ε)}(L_{n}) for all μ belonging to a certain open neighbourhood of λ and for all sufficiently large n. But then λ cannot belong to the limiting set of the ε-pseudospectra σ^{(ε)}(L_{n}) which is a contradiction.

4 Finite sections of operator polynomials

In this section, the usefulness of Theorem 2 for establishing generalizations of (1) (as well as of similar results) to operator polynomials will be illustrated. The crucial point is that there are a lot of C^{∗}-subalgebras of F/G the structure of which is sufficiently well understood (see, e.g., [4] for algebras arising from spline approximation sequences for singular integral and Mellin operators). To mention at least one non-trivial (but well known) example, consider again the finite section method for Toeplitz operators, say with piecewise continuous coefficients (where piecewise continuity of a function f on the unit circle means that both one-sided limits of f exist and are finite at every point of 𝕋). That is, the algebra F is specified accordingly by choosing C_{n} = ℂ^{n×n}, and one considers the smallest closed subalgebra S(P C) of F which contains all finite section sequences (T_{n}(f))_{n≥1} with f being piecewise continuous. The algebra S(P C) contains the ideal G (see [2]), thus it makes sense to consider the quotient algebra S(P C)/G.

One property which belongs to every sequence (A_{n}) in S(P C) is the existence of two associated strong limits: If P_{n} and R_{n} stand for the projection and reflection operators

P_{n} : l^{2} → l^{2}, (x_{1}, x_{2}, …) ↦ (x_{1}, x_{2}, …, x_{n}, 0, 0, …),
R_{n} : l^{2} → l^{2}, (x_{1}, x_{2}, …) ↦ (x_{n}, x_{n−1}, …, x_{1}, 0, 0, …),

respectively, then the strong limits

W (A_{n}) := s-lim_{n→∞}A_{n}P_{n} and W̃ (A_{n}) := s-lim_{n→∞}R_{n}A_{n}R_{n}

exist for every sequence (A_{n}) ∈ S(P C). This is evident for the generating sequences (T_{n}(f)) = (P_{n}T (f)P_{n}) of S(P C) since P_{n} → I strongly as n → ∞ and since R_{n}T_{n}(f)R_{n} = P_{n}T (f̃)P_{n} with f̃(t) = f(1/t), and follows then easily for arbitrary sequences by taking into account that the mappings W and W̃ are continuous algebra homomorphisms. Clearly, the operators W (A_{n}) and W̃ (A_{n}) depend on the coset (A_{n}) + G only. The following result due to Silbermann [11] describes the quotient algebra S(P C)/G completely by means of the homomorphisms W, W̃ : S(P C)/G → L(l^{2}).

Theorem 4 The mapping

smb : (A_{n}) + G ↦ (W (A_{n}), W̃ (A_{n}))

is a ^{∗}-isomorphism from the C^{∗}-algebra S(P C)/G into the C^{∗}-algebra L(l^{2}) × L(l^{2}).

Let now f_{0}, f_{1}, …, f_{m} be piecewise continuous functions and consider the matrix resp. operator polynomials

L_{n}(λ) := T_{n}(f_{0}) + T_{n}(f_{1})λ + … + T_{n}(f_{m})λ^{m},
P (λ) := T (f_{0}) + T (f_{1})λ + … + T (f_{m})λ^{m}.

Theorem 5 Let m > 0 and the functions f_{j} as well as the polynomials L_{n} and P be as above, and suppose the Toeplitz operator T (f_{m}) to be invertible. Then, for every ε > 0,

lim σ^{(ε)}(L_{n}) = σ^{(ε)}(P).

Proof. Let L stand for the polynomial

L(λ) := ((T_{n}(f_{0})) + G) + ((T_{n}(f_{1})) + G)λ + … + ((T_{n}(f_{m})) + G)λ^{m}.

Due to Theorem 2,

lim σ^{(ε)}(L_{n}) = σ_{F/G}^{(ε)}(L)

and, since C^{∗}-algebras are inverse closed with respect to ε-pseudo-invertibility, one even has

lim σ^{(ε)}(L_{n}) = σ_{S(P C)/G}^{(ε)}(L).

Now apply the ^{∗}-isomorphism smb from Theorem 4 to obtain

lim σ^{(ε)}(L_{n}) = σ_{L(l^{2})×L(l^{2})}^{(ε)}(smb L).

For the image of L under the homomorphisms W and W̃ one easily finds

W (L(λ)) = T (f_{0}) + T (f_{1})λ + … + T (f_{m})λ^{m} = P (λ),
W̃ (L(λ)) = T (f̃_{0}) + T (f̃_{1})λ + … + T (f̃_{m})λ^{m} =: P̃(λ)

where again f̃(t) := f(1/t), which yields

lim σ^{(ε)}(L_{n}) = σ_{L(l^{2})}^{(ε)}(P) ∪ σ_{L(l^{2})}^{(ε)}(P̃).

Finally, if C refers to the operator of conjugation,

C : l^{2} → l^{2}, (x_{1}, x_{2}, …) ↦ (\overline{x_{1}}, \overline{x_{2}}, …)

then it is elementary to check that the identity

T (f̃) = CT (f)^{∗}C

holds even for arbitrary f ∈ L^{∞}(𝕋). Thus,

P̃(λ) = CT (f_{0})^{∗}C + CT (f_{1})^{∗}Cλ + … + CT (f_{m})^{∗}Cλ^{m}
= C(T (f_{0}) + T (f_{1})λ + … + T (f_{m})λ^{m})^{∗}C = CP (λ)C

and, hence, σ^{(ε)}(P) = σ^{(ε)}(P̃).

Observe that an analogous assertion holds, and the same proof also applies, to operator polynomials whose coefficients are arbitrary sequences in S(P C). Moreover, one can derive analogous results in any case where a subalgebra of F/G is sufficiently well understood. Finally, the result of Theorem 5 can be refined by considering besides the partial limiting set lim σ^{(ε)}(L_{n}) the corresponding uniform limiting set Lim σ^{(ε)}(L_{n}). By definition, the uniform limiting set LimM_{n} of a sequence of subsets M_{n} ⊆ ℂ consists of all (uniform) limits of sequences (m_{n}) of points m_{n} ∈ M_{n}.

Theorem 6 Let the hypothesis be as in the preceding theorem. Then

lim σ^{(ε)}(L_{n}) = Lim σ^{(ε)}(L_{n}).

The point is that the algebra S(P C) is fractal in the sense of [9] which roughly speaking means that, given a subsequence (A_{n_{k}})_{k≥1} of a sequence (A_{n}) ∈ S(P C), it is possible to reconstruct the complete sequence (A_{n}) up to sequences in the ideal G. In the present situation, this fractality is a simple consequence of Theorem 4: If a subsequence (A_{n_{k}}) is known, then it is already possible to compute the strong limits W (A_{n}) and W̃ (A_{n}), which then determine the coset (A_{n}) + G uniquely. It is easy to check that it is this fractality which makes several limiting processes (such as that one considered above) uniform. For more details see [9] and also the textbook [5] which will be finished soon.

References

[1] A. Böttcher, Pseudospectra and singular values of large convolution operators. – Journal of Integral Equations and Applications 6(1994), 3, 267 –301.

[2] A. Böttcher, B. Silbermann, The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols. – Math. Nachr. 110(1983), 279 – 291.

[3] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. – Springer Verlag (to appear).

[4] R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations. – Birkhäuser Verlag, Basel, Boston, Berlin 1995.

[5] R. Hagen, S. Roch, B. Silbermann, C^{∗}-algebras and Numerical Analysis. – Textbook in preparation.

[6] H. Landau, On Szegö’s eigenvalue distribution theorem and non-Hermitian kernels. – J. d’Analyse Math. 28(1975), 335 – 357.

[7] H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory. – Quaterly Appl. Math., April 1977, 165 – 171.

[8] L. Reichel, L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. – Linear Algebra Appl. 162(1992), 153 – 185.

[9] S. Roch, B. Silbermann, C^{∗}-algebra techniques in numerical analysis. – Journal of Operator Theory 35(1996), 2, 241 – 280.

[10] L. Rodman, An Introduction to Operator Polynomials. – Operator Theory: Advances and Applications, Vol 38, Birkhäuser Verlag, Basel, Boston, Berlin 1989.

[11] B. Silbermann, Symbol constructions in numerical analysis. – In: Petkov, Lazarov (Eds.): Integral equations and inverse problems, Pitman Research Notes in Mathematics Series, Vol. 235, 1991.

[12] L. N. Trefethen, Pseudospectra of matrices. – In: D. F. Griffiths, G. A. Watson (Eds.), Numerical Analysis 1991, Longman 1992, 234 – 266.

[13] L. N. Trefethen, Non-normal matrices and pseudospectra. – Monograph in preparation.

Author’s address:

Technische Universität Darmstadt Fachbereich Mathematik Schlossgartenstrasse 7 D - 64289 Darmstadt FRG E-mail: roch@mathematik.tu-darmstadt.de