A local spectral theory for operators. III. Resolvents, spectral sets and similarity

Stampfli, J. G.

doi:10.1090/S0002-9947-1972-0295114-0

A local spectral theory for operators. III. Resolvents, spectral sets and similarity
HTML articles powered by AMS MathViewer

by J. G. Stampfli PDF

Trans. Amer. Math. Soc. 168 (1972), 133-151 Request permission

Abstract:

Let $T$ be a bounded linear operator on a Hilbert space and assume $T$ has thin spectrum. When is $T$ similar to a normal operator? This problem is studied in a variety of situations and sufficient conditions are given in terms of characteristic functions, resolvents, spectral sets, and spectral resolutions. By contrast, the question “When is $T$ normal?” has a relatively simple answer since in that case a necessary and sufficient condition can be given in terms of the resolvent alone.

References

Robert G. Bartle, Spectral localization of operators in Banach spaces, Math. Ann. 153 (1964), 261–269. MR 164243, DOI 10.1007/BF01362416

A strange dilation theorem

12

Chandler Davis and Ciprian Foiaş, Operators with bounded characteristic function and their $J$-unitary dilation, Acta Sci. Math. (Szeged) 32 (1971), 127–139. MR 313845
William F. Donoghue Jr., On a problem of Nieminen, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 31–33. MR 152892, DOI 10.1007/BF02684290
Nelson Dunford, Spectral theory. II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614. MR 51435, DOI 10.2140/pjm.1952.2.559
Nelson Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321–354. MR 63563, DOI 10.2140/pjm.1954.4.321
Nelson Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217–274. MR 104865, DOI 10.1090/S0002-9904-1958-10219-0
P. L. Duren, H. S. Shapiro, and A. L. Shields, Singular measures and domains not of Smirnov type, Duke Math. J. 33 (1966), 247–254. MR 199359, DOI 10.1215/S0012-7094-66-03328-X
S. R. Foguel, A counterexample to a problem of Sz.-Nagy, Proc. Amer. Math. Soc. 15 (1964), 788–790. MR 165362, DOI 10.1090/S0002-9939-1964-0165362-X

Some applications of spectral sets

Harmonic-spectral measure

10

61

22

Ciprian Foiaş, The class $C_{0}$ in the theory of decomposable operators, Rev. Roumaine Math. Pures Appl. 14 (1969), 1433–1440. MR 290161
Roger Godement, Théorèmes taubériens et théorie spectrale, Ann. Sci. École Norm. Sup. (3) 64 (1947), 119–138 (1948) (French). MR 0023242, DOI 10.24033/asens.945
I. C. Gohberg and M. G. Kreĭn, On a description of contraction operators similar to unitary ones, Funkcional. Anal. i Priložen. 1 (1967), 38–60 (Russian). MR 0213907
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
G. K. Leaf, A spectral theory for a class of linear operators, Pacific J. Math. 13 (1963), 141–155. MR 150593, DOI 10.2140/pjm.1963.13.141
Edgar R. Lorch, The integral representation of weakly almost-periodic transformations in reflexive vector spaces, Trans. Amer. Math. Soc. 49 (1941), 18–40. MR 3465, DOI 10.1090/S0002-9947-1941-0003465-X
V. I. Macaev, A class of completely continuous operators, Dokl. Akad. Nauk SSSR 139 (1961), 548–551 (Russian). MR 0131769
A. S. Markus, Certain criteria for the completeness of a system of root-vectors of a linear operator in a Banach space, Mat. Sb. (N.S.) 70 (112) (1966), 526–561 (Russian). MR 0216316
Charles A. McCarthy, The nilpotent part of a spectral operator, Pacific J. Math. 9 (1959), 1223–1231. MR 108728, DOI 10.2140/pjm.1959.9.1223
C. A. McCarthy and J. Schwartz, On the norm of a finite Boolean algebra of projections, and applications to theorems of Kreiss and Morton, Comm. Pure Appl. Math. 18 (1965), 191–201. MR 180867, DOI 10.1002/cpa.3160180118
Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
George Piranian, Two monotonic, singular, uniformly almost smooth functions, Duke Math. J. 33 (1966), 255–262. MR 199320
C. R. Putnam, Eigenvalues and boundary spectra, Illinois J. Math. 12 (1968), 278–282. MR 226440, DOI 10.1215/ijm/1256054217
L. A. Sahnovič, Nonunitary operators with absolutely continuous spectrum, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 52–64 (Russian). MR 0244793
Jack Schwartz, Subdiagonalization of operators in Hilbert space with compact imaginary part, Comm. Pure Appl. Math. 15 (1962), 159–172. MR 144212, DOI 10.1002/cpa.3160150205
Harold S. Shapiro, Remarks concerning domains of Smirnov type, Michigan Math. J. 13 (1966), 341–348. MR 204633
J. G. Stampfli, Analytic extensions and spectral localization, J. Math. Mech. 16 (1966), 287–296. MR 0196500, DOI 10.1512/iumj.1967.16.16019
Joseph G. Stampfli, Minimal range theorems for operators with thin spectra, Pacific J. Math. 23 (1967), 601–612. MR 229077, DOI 10.2140/pjm.1967.23.601
J. G. Stampfli, A local spectral theory for operators, J. Functional Analysis 4 (1969), 1–10. MR 0243376, DOI 10.1016/0022-1236(69)90018-4
J. G. Stampfli, A local spectral theory for operators. II, Bull. Amer. Math. Soc. 75 (1969), 803–806. MR 244794, DOI 10.1090/S0002-9904-1969-12301-3
Béla de Sz. Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Univ. Szeged. Sect. Sci. Math. 11 (1947), 152–157. MR 22309
Béla Sz.-Nagy and Ciprian Foiaş, Sur les contractions de l’espace de Hilbert. VII. Triangulations canoniques. Fonction minimum, Acta Sci. Math. (Szeged) 25 (1964), 12–37 (French). MR 170216
Béla Sz.-Nagy and Ciprian Foiaş, Sur les contractions de l’espace de Hilert. VIII. Fonctions caractéristiques. Modèles fonctionnels, Acta Sci. Math. (Szeged) 25 (1964), 38–71 (French). MR 172122
Béla Sz.-Nagy and Ciprian Foiaş, Sur les contractions de l’espace de Hilbert. X. Contractions similaires à des transformations unitaires, Acta Sci. Math. (Szeged) 26 (1965), 79–91 (French). MR 201979
Béla Sz.-Nagy and Ciprian Foiaş, On certain classes of power-bounded operators in Hilbert space, Acta Sci. Math. (Szeged) 27 (1966), 17–25. MR 198254

Similitude des opérateurs de classe ${C_p}$ à des contractions

264

35

Analyse harmonique des opérateurs de l’espace de Hilbert

37

M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
Stefan Warschawski, Über das Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung, Math. Z. 35 (1932), no. 1, 321–456 (German). MR 1545302, DOI 10.1007/BF01186562
John Wermer, The existence of invariant subspaces, Duke Math. J. 19 (1952), 615–622. MR 50799
František Wolf, Operators in Banach space which admit a generalized spectral decomposition, Nederl. Akad. Wetensch. Proc. Ser. A. 60 = Indag. Math. 19 (1957), 302–311. MR 0090782, DOI 10.1016/S1385-7258(57)50041-3
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587