OPERATOR FUNCTIONS AND LOCALIZATION OF SPECTRA
VECTOR AND MATRIX NORMS CLASSES OF MATRICES EIGENVALUES OF MATRICES MATRIX-VALUED FUNCTIONS CONTOUR INTEGRALS ALGEBRAIC EQUATIONS THE TRIANGULAR REPRESENTATION OF MATRICES NOTES REFERENCES 2 NORMS OF MATRIX-VALUED FUNCTIONS ESTIMATES FOR THE EUCLIDEAN NORM OF THE RESOLVENT EXAMPLES RELATIONS FOR EIGENVALUES AN AUXILIARY INEQUALITY EUCLIDEAN NORMS OF POWERS OF NILPOTENT MATRICES PROOF OF THEOREM 2 1 1 ESTIMATES FOR THE NORM OF ANALYTIC MATRIX-VALUED FUNCTIONS PROOF OF THEOREM 2 7 1 THE FIRST MULTIPLICATIVE REPRESENTATION OF THE RESOLVENT 2 10 THE SECOND MULTIPLICATIVE REPRESENTATION OF THE RESOLVENT 2 11 THE FIRST RELATION BETWEEN DETERMINANTS AND RESOLVENTS 2 12 THE SECOND RELATION BETWEEN DETERMINANTS AND RESOLVENTS 2 14 AN ADDITIONAL ESTIMATE FOR RESOLVENTS 2 15 NOTES 3 INVERTIBILITY OF FINITE MATRICES PRELIMINARY RESULTS LP-NORMS OF POWERS OF NILPOTENT MATRICES INVERTIBILITY IN THE NORM ? ?P (1 < P < ?) INVERTIBILITY IN THE NORM ? ?? PROOF OF THEOREM 3 4 1 POSITIVE INVERTIBILITY OF MATRICES POSITIVE MATRIX-VALUED FUNCTIONS 4 LOCALIZATION OF EIGENVALUES OF FINITE MATRICES DE?NITIONS AND PRELIMINARIES PERTURBATIONS OF MULTIPLICITIES AND MATCHING DISTANCE PERTURBATIONS OF EIGENVECTORS AND EIGENPROJECTORS PERTURBATIONS OF MATRICES IN THE EUCLIDEAN NORM UPPER BOUNDS FOR EIGENVALUES IN TERMS OF THE EUCLIDEAN NORM LOWER BOUNDS FOR THE SPECTRAL RADIUS ADDITIONAL BOUNDS FOR EIGENVALUES PROOF OF THEOREM 4 7 1 5 BLOCK MATRICES AND ?-TRIANGULAR MATRICES INVERTIBILITY OF BLOCK MATRICES ?-TRIANGULAR MATRICES MULTIPLICATIVE REPRESENTATION OF RESOLVENTS OF ?-TRIANGULAR OPERATORS INVERTIBILITY WITH RESPECT TO A CHAIN OF PROJECTORS PROOF OF THEOREM 5 1 1 6 NORM ESTIMATES FOR FUNCTIONS OF COMPACT OPERATORS IN A HILBERT SPACE BOUNDED OPERATORS IN A HILBERT SPACE COMPACT OPERATORS IN A HILBERT SPACE TRIANGULAR REPRESENTATIONS OF COMPACT OPERATORS RESOLVENTS OF HILBERT-SCHMIDT OPERATORS EQUALITIES FOR EIGENVALUES OF A HILBERT-SCHMIDT OPERATOR OPERATORS HAVING HILBERT-SCHMIDT POWERS RESOLVENTS OF NEUMANN-SCHATTEN OPERATORS PROOFS OF THEOREMS 6 7 1 AND 6 7 3 REGULAR FUNCTIONS OF HILBERT-SCHMIDT OPERATORS 6 10 A RELATION BETWEEN DETERMINANTS AND RESOLVENTS 6 11 NOTES 7 FUNCTIONS OF NON-COMPACT OPERATORS TERMINOLOGY P-TRIANGULAR OPERATORS SOME PROPERTIES OF VOLTERRA OPERATORS POWERS OF VOLTERRA OPERATORS RESOLVENTS OF P-TRIANGULAR OPERATORS TRIANGULAR REPRESENTATIONS OF QUASI-HERMITIAN OPERATORS RESOLVENTS OF OPERATORS WITH HILBERT-SCHMIDT HERMITIAN COMPONENTS OPERATORS WITH THE PROPERTY AP ?(A?)P ?C2 RESOLVENTS OF OPERATORS WITH NEUMANN - SCHATTEN HERMITIAN COMPONENTS 7 10 REGULAR FUNCTIONS OF BOUNDED QUASI-HERMITIAN OPERATORS 7 11 PROOF OF THEOREM 7 10 1 7 12 REGULAR FUNCTIONS OF UNBOUNDED OPERATORS 7 13 TRIANGULAR REPRESENTATIONS OF REGULAR FUNCTIONS 7 14 TRIANGULAR REPRESENTATIONS OF QUASIUNITARY OPERATORS 7 15 RESOLVENTS AND ANALYTIC FUNCTIONS OF QUASIUNITARY OPERATORS 7 16 NOTES 8 BOUNDED PERTURBATIONS OF NONSELFADJOINT OPERATORS INVERTIBILITY OF BOUNDEDLY PERTURBED P-TRIANGULAR OPERATORS RESOLVENTS OF BOUNDEDLY PERTURBED P-TRIANGULAR OPERATORS ROOTS OF SCALAR EQUATIONS SPECTRAL VARIATIONS PERTURBATIONS OF COMPACT OPERATORS PERTURBATIONS OF OPERATORS WITH COMPACT HERMITIAN COMPONENTS 9 SPECTRUM LOCALIZATION OF NONSELF-ADJOINT OPERATORS INVERTIBILITY CONDITIONS PROOFS OF THEOREMS 9 1 1 AND 9 1 3 RESOLVENTS OF QUASINORMAL OPERATORS UPPER BOUNDS FOR SPECTRA INNER BOUNDS FOR SPECTRA BOUNDS FOR SPECTRA OF HILBERT-SCHMIDT OPERATORS VON NEUMANN-SCHATTEN OPERATORS OPERATORS WITH HILBERT-SCHMIDT HERMITIAN COMPONENTS OPERATORS WITH NEUMANN-SCHATTEN HERMITIAN COMPONENTS 9 10 NOTES 10 MULTIPLICATIVE REPRESENTATIONS OF RESOLVENTS 10 1 OPERATORS WITH FINITE CHAINS OF INVARIANT PROJECTORS 10 2 COMPLETE COMPACT OPERATORS 10 3 THE SECOND REPRESENTATION FOR RESOLVENTS OF COMPLETE COMPACT OPERATORS 10 4 OPERATORS WITH COMPACT INVERSE ONES 10 5 MULTIPLICATIVE INTEGRALS 10 6 RESOLVENTS OF VOLTERRA OPERATORS 10 7 RESOLVENTS OF P-TRIANGULAR OPERATORS 10 8 NOTES 11 RELATIVELY P -TRIANGULAR OPERATORS 11 1 DE?NITIONS AND PRELIMINARIES 11 2 RESOLVENTS OF RELATIVELY P-TRIANGULAR OPERATORS 11 3 INVERTIBILITY OF PERTURBED RPTO 11 4 RESOLVENTS OF PERTURBED RPTO 11 5 RELATIVE SPECTRAL VARIATIONS 11 6 OPERATORS WITH VON NEUMANN-SCHATTEN RELATIVELY NILPOTENT PARTS 11 7 NOTES 12 RELATIVELY COMPACT PERTURBATIONS OF NORMAL OPERATORS 12 1 INVERTIBILITY CONDITIONS 12 2 ESTIMATES FOR RESOLVENTS 12 3 BOUNDS FOR THE SPECTRUM 12 4 OPERATORS WITH RELATIVELY VON NEUMANN - SCHATTEN O?-DIAGONAL PARTS 12 5 NOTES 13 IN?NITE MATRICES IN HILBERT SPACES AND DI?ERENTIAL OPERATORS 13 1 MATRICES WITH COMPACT O?DIAGONALS 13 2 MATRICES WITH RELATIVELY COMPACT O?-DIAGONALS 13 3 A NONSELFADJOINT DI?ERENTIAL OPERATOR 13 4 INTEGRO-DI?ERENTIAL OPERATORS 13 5 NOTES 14 1 SCALAR INTEGRAL OPERATORS 14 2 MATRIX INTEGRAL OPERATORS WITH RELATIVELY SMALL KERNELS 14 3 PERTURBATIONS OF MATRIX CONVOLUTIONS 14 4 NOTES 15 OPERATOR MATRICES 15 1 INVERTIBILITY CONDITIONS 15 2 BOUNDS FOR THE SPECTRUM 15 3 OPERATOR MATRICES WITH NORMAL ENTRIES 15 4 OPERATOR MATRICES WITH BOUNDED O?DIAGONAL ENTRIES 15 5 OPERATOR MATRICES WITH HILBERT-SCHMIDT DIAGONAL OPERATORS 15 6 EXAMPLE 15 7 NOTES 16 HILLE - TAMARKIN INTEGRAL OPERATORS 16 1 INVERTIBILITY CONDITIONS 16 2 PRELIMINARIES 16 3 POWERS OF VOLTERRA OPERATORS 16 4 SPECTRAL RADIUS OF A HILLE - TAMARKIN OPERATOR 16 5 NONNEGATIVE INVERTIBILITY 16 6 APPLICATIONS 16 7 NOTES 17 INTEGRAL OPERATORS IN SPACE L? 17 1 INVERTIBILITY CONDITIONS 17 2 PROOF OF THEOREM 17 1 1 17 3 THE SPECTRAL RADIUS 17 4 NONNEGATIVE INVERTIBILITY 17 5 APPLICATIONS 17 6 NOTES 18 HILLE - TAMARKIN MATRICES 18 1 INVERTIBILITY CONDITIONS 18 2 PROOF OF THEOREM 18 1 1 18 3 LOCALIZATION OF THE SPECTRUM 18 4 NOTES 19 ZEROS OF ENTIRE FUNCTIONS 19 1 PERTURBATIONS OF ZEROS 19 2 PROOF OF THEOREM 19 1 2 19 3 BOUNDS FOR SUMS OF ZEROS 19 4 APPLICATIONS OF THEOREM 19 3 1 19 5 NOTES |XK|P]1/P (1 ?P < ?) AND ?X??= MAX K=1,,N |XK| FOR AN X = (XK) ?CN THE NORM ?X?2 IS CALLED THE EUCLIDEAN NORM THROUGHOUT THIS CHAPTER ?X?MEANS AN ARBITRARY NORM OF A VECTOR X WE WILL USE THE FOLLOWING MATRIX NORMS: THE OPERATOR NORM AND THE FROBENIUS ARE THE IMAGINARY HERMITIAN COMPONENT AND THE REAL HERMITIAN ONE OF A, RESPECTIVELY A MATRIX A IS DISSIPATIVE IF ITS REAL HERMITIAN COMPONENT IS NEGATIVE DE?NITE BY A?1 THE MATRIX INVERSE TO A IS DENOTED: AA?1 = A?1A = I 1 3 EIGENVALUES OF MATRICES (? ??K(A)) IS SAID TO BE THE CHARACTERISTIC POLYNOMIAL OF A ALL THE EIGENVALUES OF A ARE THE ROOTS OF ITS CHARACTERISTIC POLYNOMIAL THE ALGEBRAIC MULTIPLICITY OF AN EIGENVALUE OF A COINCIDES WITH THE MULTIPLICITY OF THE CORRESPONDING ROOT OF THE CHARACTERISTIC POLYNOMIAL A POLYNOMIAL IS SAID TO BE A HURWITZ ONE IF ALL ITS ROOTS LIE IN THE OPEN LEFT HALF-PLANE THUS, THE CHARACTERISTIC POLYNOMIAL OF A HURWITZ MATRIX IS A HURWITZ POLYNOMIAL AN ? ? ? ? THEN F(A) = ? ? ? ? F(A1) F(A2) F(AN) ? ? ? ? EXAMPLE 1 4 2 IF A MATRIX J IS AN N × N-JORDAN BLOCK: 1 5 CONTOUR INTEGRALS ? ? ? ? ? ? ? ? ? ? , THEN F(J) = ? ? ? ? ? ? ? ? ? ? ? F(?0) F ?(?0) 1! F (N?1)(?0) (N?1)! F(?0) F(?0) F ?(?0) 1! CONTOUR INTEGRALS LEMMA 1 5 1 LET M0 BE THE CLOSED CONVEX HULL OF POINTS X0, X1, , XN ?C AND LET A SCALAR-VALUED FUNCTION F BE REGULAR ON A NEIGHBORHOOD D1 OF M0 IN ADDITION, LET ? ?D1 BE A JORDAN CLOSED CONTOUR SURROUNDING THE POINTS X0, X1, , XN THEN | |F (N)(?)| PROOF: FIRST, LET ALL THE POINTS BE DISTINCT: XJ ?= XK FOR J ?= K (J, K = 0, , N), AND LET DF(X0, X1, , XN) BE A DIVIDED DI?ERENCE OF FUNCTION F AT POINTS X0, X1, , XN THE DIVIDED DI?ERENCE ADMITS THE REPRESENTATION DF(X0, X1, , XN) = |F (N)(?)| (GEL’FOND, 1967, FORMULA (49)) COMBINING THAT INEQUALITY WITH RELATION (5 1), WE ARRIVE AT THE REQUIRED RESULT IF XJ = XK FOR SOME J ?= K, THEN THE CLAIMED INEQUALITY CAN BE OBTAINED BY SMALL PERTURBATIONS AND THE PREVIOUS REASON- INGS 1 PRELIMINARIES LEMMA 1 5 2 LET X0 ?X1 ? ?XN BE REAL POINTS AND LET A FUNCTION F BE REGULAR ON A NEIGHBORHOOD D1 OF THE SEGMENT [X0, XN] IN ADDITION, LET ? ?D1 BE A JORDAN CLOSED CONTOUR SURROUNDING [X0, XN] THEN THERE IS A POINT ? ?[X0, XN], SUCH THAT THE EQUALITY 2?I ? F(?)D? (? ?X0)(? ?XN) = N! F (N)(?) IS TRUE PROOF: FIRST SUPPOSE THAT ALL THE POINTS ARE DISTINCT: X0 < X1 < < XN THEN THE DIVIDED DI?ERENCE DF(X0, X1, , XN) OF F IN THE POINTS X0, X1, , XN ADMITS THE REPRESENTATION DF(X0, X1, , XN) = N!F (N)(?) WITH SOME POINT ? ?[X0, XN] (GEL’FOND, 1967, FORMULA (43)), (OSTROWSKI, 1973, PAGE 5 ) COMBINING THAT EQUALITY WITH REPRESENTATION (5 1), WE ARRIVE AT THE REQUIRED RESULT IF XJ = XK FOR SOME J ?= K, THEN THE CLAIMED INEQUALITY CAN BE OBTAINED BY SMALL PERTURBATIONS AND THE PREVIOUS REASONINGS P(1) IN THIS CASE SUBSTITUTING THIS INEQUALITY IN (6 1), WE GET (6 3) 1 7 THE TRIANGULAR REPRESENTATION AJKEJ WITH AJK = (AEK, EJ) (J = 1, , N), (7 2) WHERE ( , ) IS THE SCALAR PRODUCT THIS BASIS IS CALLED SCHUR’S BASIS OF THE OPERATOR A IN ADDITION, AJJ = ?J(A), WHERE ?J(A) ARE THE EIGENVALUES OF A ACCORDING TO (7 2), A = D + V (7 3) WITH A NORMAL (DIAGONAL) OPERATOR D DE?NED BY DEJ = ?J(A)EJ (J = 1, , N) ( , EK)EK (J = 1, , N), P0 = 0 THEN 0 ?P1CN ? ?PNCN = CN MOREOVER, APK = PKAPK; V PK?1 = PKV PK; DPK = DPK (K = 1, , N) (7 4) SO A, V AND D HAVE THE SAME CHAIN OF INVARIANT SUBSPACES LEMMA 1 7 1 LET Q, V ?B(CN) AND LET V BE A NILPOTENT OPERATOR SUPPOSE THAT ALL THE INVARIANT SUBSPACES OF V AND OF Q ARE THE SAME THEN V Q AND QV ARE NILPOTENT OPERATORS PROOF: SINCE ALL THE INVARIANT SUBSPACES OF V AND Q ARE THE SAME, THESE OPERATORS HAVE THE SAME BASIS OF THE TRIANGULAR REPRESENTATION TAKING INTO ACCOUNT THAT THE DIAGONAL ENTRIES OF V ARE EQUAL TO ZERO, WE EASILY DETERMINE THAT THE DIAGONAL ENTRIES OF QV AND V Q ARE EQUAL TO ZERO THIS PROVES THE REQUIRED RESULT NOTES THIS BOOK PRESUPPOSES A KNOWLEDGE OF BASIC MATRIX THEORY, FOR WHICH THERE ARE GOOD INTRODUCTORY TEXTS THE BOOKS (BELLMAN, 1970), (GANTMAHER, 1967), ( MARCUS AND MINC, 1964) ARE CLASSICAL FOR MORE DETAILS ABOUT THE NOTIONS PRESENTED IN SECTIONS 1 1-1 4 ALSO SEE (COLLATZ, 1966) AND (STEWART AND SUN, 1990) ESTIMATES FOR ROOTS OF ALGEBRAIC EQUATIONS SIMILAR TO COROLLARY 1 6 2 CAN BE FOUND IN (OSTROWSKI, 1973, PAGE 277) REFERENCES [1] BELLMAN, R E (1970) INTRODUCTION TO MATRIX ANALYSIS MCGRAW-HILL, NEW YORK 1 8 NOTES