© Hindawi Publishing Corp. LOCAL SPECTRAL THEORY FOR 2×2 OPERATOR MATRICES

We discuss the spectral properties of the operator MC ∈ (X ⊕ Y) defined by MC := AC 0 B ,w hereA ∈ (X), B ∈ (Y ), C ∈ (Y , X) ,a ndX, Y are complex Banach spaces. We prove that (SA∗ ∩ SB ) ∪ σ( MC ) = σ( A)∪ σ( B)for all C ∈ (Y , X). This allows us to give a partial positive answer to Question 3 of Du and Jin (1994) and generalizations of some results of Houimdi and Zguitti (2000). Some applications to the similarity problem are also given.


Introduction.
Let X and Y be complex Banach spaces and let ᏸ(X), ᏸ(Y ), and ᏸ(Y , X) be the algebras of all continuous linear operators on X, Y , and from Y to X, respectively.For T ∈ ᏸ(X), we denote by σ (T ) its spectrum, σ p (T ) its point spectrum, T * its adjoint operator, and R T its resolvent map.Let x ∈ X and λ 0 ∈ C, we say that λ 0 is in the local resolvent of T at x, denoted by ρ T (x), if the equation admits an analytic solution in a neighborhood of λ 0 .The set σ T (x) = C\ρ T (x) is called the local spectrum of T at x.If for every x ∈ X any two solutions of (1.1) agree on their common domain, T is said to have the single-valued extension property (SVEP).It is obvious that T has the SVEP if and only if the zero function is the only analytic function which satisfies (T − λ)f (λ) = 0.
For T ∈ ᏸ(X) and F a closed set of C, denote by the set X T (F ) := {x ∈ X, σ T (x) ⊂ F } the analytic spectral space.The analytic residuum S T is the set of λ 0 ∈ C for which there exist a neighborhood G λ 0 and f : G λ 0 → X a nonzero analytic function such that (T − λ)f (λ) = 0 for all λ ∈ G λ 0 .We say that the operator T has the Dunford condition C (DCC) if X T (F ) is closed whenever F is closed.It is clear that T has the SVEP if and only if S T = ∅.The surjective spectrum of T is given by σ su (T ) := {λ ∈ C/ T−λ is not surjective}.It is known that σ su (T ) = ∪ x∈X σ T (x) and that σ (T ) = S T ∪ σ su (T ) (see [6,7,8]).In particular, if T has the SVEP, we obtain that σ (T ) = σ su (T ).A complete study of basic notions of local spectral theory can be found in [1,4].
and X, Y are complex Banach spaces.In [2], a number of natural questions concerning the relation between the spectrums of these operators have been considered.Our interest in this paper is to develop some new conditions under which we have the equality σ (M C ) = σ (A)∪σ (B).Houimdi and Zguitti [3] give a positive answer to this question when the operator B has the SVEP.
Our main result gives a partial answer to [2, Question 3], and generalizes results from [3].At the end, we give a generalization of [3, Proposition 3.2] related to the DCC for the operator M C and some applications to the similarity of orbits.
We collect in the following proposition some useful spectral properties of the operator M C from [2,3], that can be also obtained easily.

Spectral theory of the operator M C .
We study in the sequel spectral theory of M C , we refine the inclusions given in Proposition 1.1, and provide some properties of local spectrum.

.11)
For the converse, we obtain by Proposition 2.3 that We are now in a position to derive a generalization of [3, Theorem 2.1].
Theorem 2.5.For given operators A, B, and C, (2.15) Proof.In the first step, we prove that (2.16) By Corollary 2.4 and Proposition 1.1(3), we obtain that (2.17) The converse is clear since the inclusion σ (M C ) ⊂ σ (A)∪ σ (B) is always true.
In the second step, remark that (2.18) Thus, in the same way, we obtain that

.19)
Hence the theorem is proved.
We give in what follows a new condition under which we have the desired equality.

Applications.
In this section, we give some necessary conditions such that M C has the SVEP and provide some results on similarity orbits and the range of generalized derivation using Theorem 2.
Proof.Let (x n ) be a sequence of elements of X A (F ) which converges to x ∈ X.By Proposition 2.3, we have We derive that Our second application is related to the similarity problem.Let A, B, and M C be as above and consider δ A,B the derivation operator defined by δ A,B (X) = AX −XB for X ∈ ᏸ(X).Let Im(δ A,B ) be its range and denote by H 1 , H 2 , and H 3 the following classes of operators: It is obvious that The above inclusions received a lot of interest (see [9,10]).In general, any of these inclusions can be strict.We construct some classes of operators such that Remark first by Corollary 2.6 that if S A * ∩ S B = ∅, then H 3 = ᏸ(Y , X). Suppose in the sequel that X = Y and let S be the unilateral shift.We consider, respectively, the operators A := S 0 0 0 and M C := A C 0 A .By [9,Theorem 6], if M C and M O are similar, then C is a commutator, that is, C = λI + K for any λ ∈ C and K a compact operator.Hence, choosing A so that S A * ∩ S A = ∅ and considering C = λI + K for some λ ∈ C and K a compact operator, we get M C ∈ H 3 \H 2 .Moreover, for C = 0 I 0 0 , we conclude by [9] that M C ∈ H 2 \H 1 .Consequently, the inclusions in (3.6) are all strict for M C = A C 0 A with A = S 0 0 0 , C = 0 I 0 0 , and S is the unilateral shift.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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Proposition 2 . 1 .
If A, B, and C are given, then

Corollary 2 . 2 .
For given operators A, B, and C, σ (B) ⊂ σ su (A) ∪ σ M C .(2.5) To establish our main theorem, we first claim the following proposition related to the local spectrum of the operator M C , which generalizes [3, Proposition 2.1].Proposition 2.3.If A, B, and C are given,

5 .Proposition 3 . 1 .Proposition 3 . 2 .
The following result gives a partial characterization of the property of the single extension property of the operator M C and discusses the converse of[3, Proposition 3.1].For given operators A, B, and C such that the surjective spectrum of the operator A has an empty interior,S M C = ∅ ⇐⇒ S A = S B = ∅.(3.1)In particular M C has SVEP if and only if A and B have the SVEP.Proof.If A and B have the SVEP, so is the case of M C .(See Proposition 2.1(3).)On the other hand, the relationsS B ⊂ σ su (A) ∪ S M C , S A ⊂ S M C ⊂ S A ∪ S B (3.2)imply that S A = S B = ∅.The proof is complete.The next proposition provides a generalization of[3, Proposition 3.2].If A and B are given and F is a closed set such that S B ⊂ F , then the following assertion holds true: if there exists C