AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

Abstract Let A and 
$\tilde A$
 be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of 
$\tilde A$
 lie, if A and 
$\tilde A$
 are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of 
$\tilde A$
 , assuming that 
$\tilde A-A$
 is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.


Introduction and statement of the main result
Let H be a complex separable Hilbert space with a scalar product (·, ·), norm given by · = √ (·, ·) and unit operator I. By L(H) we denote the set of all bounded operators in H. For an operator A on H, D(A) is its domain, A * and A −1 are the adjoint and inverse operators, respectively, σ(A) is the spectrum, R z (A) = (A − zI) −1 (z σ(A)) is the resolvent, and λ j (A) (j = 1, 2, . . .) denote the eigenvalues of A taken with their multiplicities. In addition, for ω > 0, we denote by H ω := {z ∈ C : |Im z| < ω} the horizontal strip of height 2ω which is symmetric with respect to the real axis. Following [10, Section 4.1], we will say that an operator A on H is a strip-type operator of height ω (in short, A ∈ Strip(ω)) if σ(A) ⊂ H ω and sup |Im z|≥ω R z (A) < ∞ for all ω > ω. Finally, ω st (A) := inf{ω ≥ 0 : A ∈ Strip(ω)} is called the spectral height of A.
We consider the following problem. Let A andÃ be strip-type operators on H. In which strip does the spectrum ofÃ lie if ω st (A) is known andÃ and A are sufficiently 'close'? We also discuss applications of our results to matrix differential operators.
The strip-type operators form a wide class of unbounded operators in a Banach space. The important example here is the logarithm of a sectorial operator, arising in various applications (see [10,16]). The natural functional calculus for strip-type operators appears first in [2]. It is discussed in [11] in a general setting and used in [3]. The theory of strip-type operators is developed in [9,16,17] and the references given therein. For more details, see [10,Ch. 4]. To the best of our knowledge, the above-mentioned problem has not been considered in the literature, although it is important for the localisation of spectra and in various applications.
Furthermore, A is said to be a strong strip-type operator of height ω, if for any ω > ω there is an L ω such that Then Ã I ≤ q + A I and thereforeÃ is also a strip-type operator. We introduce the notation This inequality is rather rough. Below, we present a considerably sharper estimate.
To this end, note that according to (1.1), e ±iAt ≤ const. e ω st t (t ≥ 0), and thus the operators −(cI ± iA), for c ∈ R, generate the exponentially stable semigroups e −(cI±iA)t , provided c > ω st . Hence, the integral strongly converges and We are now in a position to formulate our main result, which we prove in Section 2.
By the classical Parseval-Plancherel equality [1, Theorem 5.2.1], for any x ∈ H, Hence, If A is normal, that is, AA * = A * A, then by the spectral representation (see, for instance, [12]), we easily see that Making use of Theorem 1.1, we obtain ω st (Ã) ≤ ω st (A) + q + for > 0. Hence, letting → 0, we arrive at the following result. Let us show that Theorem 1.1 is sharp. To this end, assume that K ∈ L(H) and A are self-adjoint commuting operators andÃ = A + iK. Suppose also that σ(A) and σ(K) are discrete. Then σ(Ã) consists of the eigenvalues k = 1, 2, . . .).

Spectral strips of differential operators with matrix coefficients
Let L 2 = L 2 ([0, 1], C n ) be the space of functions defined on [0, 1] with values in C n and the scalar product where (·, ·) n means the scalar product in C n . On the domain where C(x) is an n × n matrix continuously dependent on x. We consider this operator as a perturbation of the operator with a constant n × n matrix C 0 . By way of example, one can take C 0 = C(0) or Here A −Ã L 2 is the operator norm in L 2 of A −Ã and · n means the spectral matrix norm (the operator norm with respect to the Euclidean vector norm).
Letting y → x n + ω st (C 0 ), we obtain the following result. If C 0 is normal, then g(C 0 ) = 0, and with 0 0 = 1 we have ζ(s) = 1/s and thus x n = q. The following lemma gives us an estimate for x n in the case g(C 0 ) 0. PROOF. By (3.4), qζ(x n ) = 1 ≥ qζ (1). Since ζ(s) is monotonically decreasing, it follows that x n ≤ 1. Now (3.5) proves the lemma. For recent results on the spectra of differential operators see, for instance, the works [6,7,13,14,15,18,19] and the references which are given therein.