Block numerical ranges [Elektronische Ressource] / von Markus Wagenhofer

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Block Numerical Rangesvon Markus WagenhoferDissertationzur Erlangung des Grades eines Doktors derNaturwissenschaften– Dr.rer.nat. –Vorgelegt im Fachbereich 3 (Mathematik & Informatik)der Universit¨ at Bremenim Januar 2007Datum des Promotionskolloqiums: 14. Februar 2007Gutachter: Prof.Dr. Christiane Tretter (Universit¨ at Bern)Prof.Dr. Heinz Langer (TU Wien)ContentsIntroduction 31 The block numerical range of bounded operators 91.1 Aboutdecompositions.......................... 91.2 The definition and known results ....................11.3 Continuity properties of W .......................14H1.4 Connected components of W (A)17H1.5 Theblockdeterminantset........................242 The block numerical range of operator functions 272.1 The definition and simple properties ..................282.2 Spectralinclusion.............................312.3 Thenormoftheresolvent343 The block numerical range of operator polynomials 373.1 On the boundedness of W (P) .....................42H3.2 Connectedcomponentsandthenormoftheresolvent.........43.3 Thecompanionpolynomial........................464 Block-diagonalization of operators 494.1 Schur complements ............................504.2 Factorization of Schur complements ...................534.3 InvariantsubspacesandRicatiequations ...............615 Corners of block numerical ranges 655.1 Analyticperturbationsofmatrices....................665.2 Corners of W (A) belonging to W (A).................

Sujets

• Mathematics

Informations

• Publié par: universitat_bremen
• Publié le: 01 janvier 2007
• Nombre de lectures: 18
• Langue: English

Extrait

Block

Numerical

Ranges

von Markus Wagenhofer

Dissertation

zur Erlangung des Grades eines Doktors der Naturwissenschaften – Dr. rer. nat. –

Vorgelegt im Fachbereich 3 (Mathematik & Informatik) derUniversit¨atBremen im Januar 2007

Datum des Promotionskolloqiums: 14. Februar 2007

Gutachter: Prof. Dr. Christiane Tretter (Universit¨at Bern) Prof. Dr. Heinz Langer (TU Wien)

Contents

Introduction

1

2

3

4

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The block numerical range of bounded operators 1.1 About decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The definition and known results . . . . . . . . . . . . . . . . . . . . 1.3 Continuity properties ofW H . . . . . . . . . . . . . . . . . . . . . . . 1.4 Connected components ofW H (A) . . . . . . . . . . . . . . . . . . . . 1.5 The block determinant set . . . . . . . . . . . . . . . . . . . . . . . .

The block numerical range of operator functions 2.1 The definition and simple properties . . . . . . . . . . . . . . . . . . 2.2 Spectral inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The norm of the resolvent . . . . . . . . . . . . . . . . . . . . . . . .

The block numerical range of operator polynomials 3.1 On the boundedness ofW H (P) . . . . . . . . . . . . . . . . . . . . . 3.2 Connected components and the norm of the resolvent . . . . . . . . . 3.3 The companion polynomial . . . . . . . . . . . . . . . . . . . . . . . .

Block-diagonalization of operators 4.1 Schur complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Factorization of Schur complements . . . . . . . . . . . . . . . . . . . 4.3 Invariant subspaces and Riccati equations . . . . . . . . . . . . . . .

Corners of block numerical ranges 5.1 Analytic perturbations of matrices . . . . . . . . . . . . . . . . . . . . 5.2 Corners ofW H (A) belonging toW H (A. . . . . . . . . . . . . . . .) . 5.3 Corners ofW H (A) belonging toW H (A) . . . . . . . . . . . . . . . . . 5.4 Corners ofW H (F . . . . . . . . . . . . . . . . . . . . . . . . . . .) .

A Computing block numerical ranges

References

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Introduction

Linear equations involving block operator matrices play an important role in many applications. For instance, they occur as saddle point problems arising in the dis- cretization process of systems of partial differential equations with constraints in fluid dynamics or linear elasticity. An impressive list of more fields providing such problems (including references) is given in [BGL05]. One method for solving such problems is the preconditioning of the original problem, that is, its transformation into an equivalent problem with better spectral properties (see [BGL05, Section 10]). Knowledge about the spectral properties of a matrix or an operator is essential in the construction of effective preconditioners for linear systems. In the context of block operator matrices, it is additionally highly desirable to exploit as much information as possible from the block structure. However, even in the finite dimensional case very little is known about the spectral properties of block operator matrices. The numerical range of an operator is a well-known and effective tool when studying spectral properties of operators. However, it does not respect the block structure of block operator matrices, thus destroying information which could be of subsequent use. To address this problem, in [LT98] thequadratic numerical rangeof a block operator matrix A=  BA DC  (0.1) with respect to a decompositionH=H 1 ×H 2 was defined by x) W 2 (A ) :=   σ p  ( (y,x A C ,xx ) ( ( ) y B,y Dy,)  : x∈H 1 , y∈H 2 ,x=y= 1  .

It was used there as a tool to locate the spectrum of non selfadjoint block operator matricesAwith possibly unbounded entriesAandD. More properties of the quadratic numerical range, often similar to properties of thenumerical range

W(A ) := {(Af, f ) : f∈ H,f= 1},

were shown in [LMMT01], [LMT01], including the inclusions

σ p (A)⊂W 2 (A), σ(A)⊂W 2 (A), W 2 (A)⊂W(A),

estimates of the resolvent, and the fact that corners of ofA. Moreover, ifW 2 (A) consists of two connected

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W 2 (A) lie in the spectrum components, the existence

Introduction

of certainA-invariant subspaces and corresponding solutions of Riccati equations related toAwas proved; additionally, as a generalization of the well-known fact that a 2×2 matrix with two distinct eigenvalues is diagonalizable,Awas shown to be block-diagonalizable in this case. The definition of the quadratic numerical range was generalized to decompositions H=H 1  · · ×× ·H n  a block Givenin the obvious way in [Wag00] and [TW03]: operator matrixAwith respect to the decompositionH, A= ⎛ ⎜ ⎝ A A . 1 n 1 1 . . . A . . A . . 1 nn n ⎠ ⎞ ⎟ ,

itsblock numerical rangewith respect to this decomposition was defined by W H (A ) :=   σ p (A x ) : x= (x 1 , . . . , x n )∈ H,x i = 1, i= 1, . . . , n  ,

where (A 11 x 1 , x 1 ). . . A x := ⎝ ⎜ ⎛ (A n1 x 1 , x n ). . . .

(A 1n x n , x 1 ) ⎞ ⎟ . n , x n ⎠ , (A nn x)

x= (x 1 , . . . , x n )∈ H.

(0.2)

It was shown thatW H (A)⊂W H (A) ifHis a refined decomposition ofH, while the spectral inclusionσ(A)⊂W H (A) continues to hold. These two properties alone justify a further investigation of the block numerical range.

Aims It is the aim of this thesis to prove generalizations of theorems known for the nu- merical range of operators and the quadratic numerical range of 2×2 block operator matrices and, more generally, block operator functions, including •the block diagonalization of block operator matrices if the closure of the block numerical range consists ofnconnected components (which is, in some sense, a generalization of the well-known fact that a complexn×nmatrix withn distinct eigenvalues is diagonalizable), •the fact that corners of the block numerical range of a block operator matrix are contained in its spectrum, •the definition and properties of the block numerical range of block operator functions (generalizing [Tre06]), and, in particular, block operator polynomi- als.

Overview InChapter 1, the concept and main properties of block numerical ranges of bounded operators are presented. In particular, Chapter 1 contains the most im- portant results from [Wag00] and [TW03] (Section 1.2), including, e. g.,

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(1) (2)

Introduction

the spectral inclusionsσ p (A)⊂W H (A),σ(A)⊂W H (A), the resolvent estimate

(A −z) −1  ≤ (A+|z|) n−1 dist(z, W H (A)) n ,

z∈C\W H (A),

(0.3)

(3) the inclusionW H (A)⊂W H (A) for a decompositionHwhich is finer thanH. Section 1.3 addresses continuity properties of mappings connected with the block numerical range. For example, it is shown, that the closureW H of the numerical range depends continuously on the operator with respect to the Hausdorff metric onC. It is known that the block numerical range is, in contrast to the numerical range, not connected anymore. If the closure of the quadratic numerical range of a block operator matrixAconsists of two connected components, it was shown in [LMMT01] that there are invariant subspaces ofAgiven by the graph subspaces of solutions of certain Riccati equations related to the block entries ofA, and thatA allows a block diagonalization. To prove corresponding statements in the general block numerical case (Chapter 4), the results of Section 1.4, where the connected components of block numerical ranges are examined, are of major interest. In par- ticular, the notion of (strongly)H-separated connected components ofW H (A) is introduced (Definition 1.22) and a Gershgorin type condition on the block entries ofAfor strongH-separateness (Proposition 1.27) is presented. Section 1.5 Finally, shows relations between properties of the block numerical range of a block operator Aand its block determinant setD H (A) ={detA x : x i = 1, i= 1, . . . , n}with respect toH. InChapter 2the concept of the block numerical range is extended to block operator functionsF : Ω →L(H) by the definition

W H (F ) := {z∈ Ω : 0 ∈W H (F(z))},

which is in accordance with the definitionW(F) ={z∈ Ω : 0 ∈W(F(z))}of the numerical range (see [Mar88,§26.3]) and of the quadratic numerical range for 2×2 block operator matrix functions (see [Tre06]). The most important part of this chapter is Section 2.2 where, for analytic block operator functionsF, the spectral inclusionsσ p (F)⊂W H (F) andσ(F)⊂W H (F) are shown under an additional condition onFgeneralizing a well-known condition for the numerical range (see [Mar88, Equation (26.5)]), namely

0/∈W H (F(z 0 )) for somez 0 ∈Ω.

(0.4)

Moreover, in Section 2.3, an estimate of the resolvent normF −1 similar to (0.3) is proved, F −1 (z) ≤ dist(,zC γ ) ν , z∈U\C,

for certain connected componentsCofW H (F) and compact neighborhoodsUof them. This estimate, in turn, allows to give upper bounds for the lengths of Jordan

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