Skip to main content
SpringerLink
Log in

Non-self-adjoint operator algebras in Hilbert space

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature cited

  1. N. Aronzajn and C. Smith, “Invariant subspaces of completely continuous operators,” Matematika,2, No. 1, 97–102 (1958).

    Google Scholar 

  2. M. S. Brodskii, “A problem of I. M. Gel'fand,” Usp. Matem. Nauk,12, No. 2, 129–132 (1957).

    Google Scholar 

  3. M. S. Brodskii, “Triangular and Jordan Representations of Linear Operators [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  4. M. S. Brodskii, I. Ts. Gokhbert, M. G. Krein, and V. I. Mal'tsev, Trans. All-Union Mathematical Congress, 1961 [in Russian], Vol. 2, Nauka, Leningrad (1964), pp. 261–271.

    Google Scholar 

  5. T. Gamelin, Uniform Algebras [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  6. K. Hoffman, Banach Spaces of Analytic Functions [Russian translation], Izd-vo In. Lit., Moscow (1963).

    Google Scholar 

  7. I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  8. I. Ts. Gokhberg and M. G. Krein, Theory of Volterra Operators in Hilbert Space and Applications [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  9. N. Dunford and J. Schwartz, Linear Operators. II [Russian translation], Mir, Moscow (1966).

    Google Scholar 

  10. R. S. Ismagilov, “Rings of operators in a space with indefinite metric,” Dokl. Akad. Nauk SSSR,171, No. 2, 269–271 (1966).

    Google Scholar 

  11. G. É. Kiselevskii, “Invariant subspaces of dissipative Volterra operators with imaginary nuclear components,” Izv. Akad. Nauk SSSR. Ser. Matem.,32, No. 1, 3–23 (1968).

    Google Scholar 

  12. É. V. Kissin, “C*-algebras generated by dynamic systems and weighted shifts,” Dokl. Akad. Nauk SSSR,216, No. 6, 1215–1218 (1974).

    Google Scholar 

  13. B. I. Korenblyum, “Invariant subspaces of shift operators in a Hilbert weight space,” Matem. Sb.,89, No. 1, 110–137 (1972).

    Google Scholar 

  14. M. G. Krein, “Positively additive functionals in linear normed spaces,” Zap. Naukove Doslid. Inst. Matem. i Mekh. KhDU i Kharkov Matem. Tov.,14, 227–237 (1937).

    Google Scholar 

  15. V. I. Liberzon and V. S. Shul'man, “Nondegenerate operator algebras in IIk spaces with indefinite metric,” Izv. Akad. Nauk SSSR. Ser. Matem.,37, No. 3, 533–538 (1973).

    Google Scholar 

  16. V. I. Liberzon and V. S. Shul'man, “Operator-irreducible symmetric operator algebras in the II1 Pontryagin space,” Izv. Akad. Nauk SSSR. Ser. Matem.,35, No. 5, 1159–1170 (1971).

    Google Scholar 

  17. M. S. Lifshits, Operators, Oscillations, Waves. Open Systems [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  18. A. I. Loginov, “Commutative symmetric Banach operator algebras in the II1 Pontryagin space,” Dokl. Akad. Nauk SSSR,179, No. 6, 1276–1278 (1968).

    Google Scholar 

  19. A. I. Loginov, “Complete commutative symmetric operator algebras in the II1 Pontryagin space,” Matem. Sb.,84, No. 4, 575–582 (1971).

    Google Scholar 

  20. A. I. Loginov and V. S. Shul'man, “Sarason's theorem and the Radjavi-Rosenthal hypothesis,” Dokl. Akad. Nauk SSSR,205, No. 2, 284–285 (1972).

    Google Scholar 

  21. A. I. Loginov and V. S. Shul'man, “Hereditary and intermediate reflexivity of W*-algebras,” Dokl. Akad. Nauk SSSR,212, No. 4, 34–36 (1973).

    Google Scholar 

  22. A. I. Loginov and V. S. Shul'man, “Reductive operator algebras and the invariant subspace problem,” Dokl. Akad. Nauk SSSR,216, No. 1, 36–38 (1974).

    Google Scholar 

  23. V. I. Lomonosov, “Invariant subspaces of a family of operators that commute with a completely continuous operator,” Funkts. Anal. i Ego Prilozh.,7, No. 3, 55–56 (1973).

    Google Scholar 

  24. V. I. Matsaev, “A class of completely continuous operators,” Dokl. Akad. Nauk SSSR,138, No. 3, 548–551 (1961).

    Google Scholar 

  25. M. A. Naimark, Linear Representations of the Lorentz Group [in Russian], Fizmatgiz, Moscow (1958), 376 pp.

    Google Scholar 

  26. M. A. Naimark, Normed Rings [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  27. M. A. Naimark, “Commuting unitary operators in the ∏ℵ space,” Dokl. Akad. Nauk SSSR,149, No. 6, 1261–1263 (1963).

    Google Scholar 

  28. M. A. Naimark, “Commutative operator algebras in the II1 space,” Dokl. Akad. Nauk SSSR,156, No. 4, 734–737 (1964).

    Google Scholar 

  29. M. A. Naimark, “Commutative operator algebras in the II1 space,” Rev. Roum. Math. Pures Appl.,9, No. 6, 499–528 (1964).

    Google Scholar 

  30. M. A. Naimark, “Commutative operator algebras in the IIk space,” Dokl. Akad. Nauk SSSR,161, No. 4, 767–770 (1965).

    Google Scholar 

  31. M. A. Naimark, “Structure of unitary representations of locally bicompact groups in the II1 space,” Izv. Akad. Nauk SSSR. Ser. Matem.,29, No. 3, 689–770 (1965).

    Google Scholar 

  32. M. A. Naimark, “Structure of unitary representations of locally bicompact groups and symmetric representations of algebras in the Pontryagin IIk space,” Izv. Akad. Nauk SSSR. Set. Matem.,30, No. 5, 1111–1132 (1966).

    Google Scholar 

  33. M. A. Naimark and R. S. Ismagilov, “Representations of groups and algebras in spaces with indefinite metric,” in: Progress in Science. Mathematical Analysis. 1968 [in Russian], VINITI Akad. Nauk SSSR, Moscow (1969), pp. 73–105.

    Google Scholar 

  34. N. K. Nikol'skii, “Invariant subspaces of certain completely continuous operators,” Vestn. Leningr. Univ., No. 7, 68–77 (1965).

    Google Scholar 

  35. N. K. Nikol'skii, “Invariant subspaces of unitary operators,” Vestn. Leningr. Univ., No. 19, 36–43 (1966).

    Google Scholar 

  36. N. K. Nikol'skii, “Unicellularity and nonunicellularity of weighted shift operators,” Dokl. Akad. Nauk SSSR,172, No. 2, 287–290 (1967).

    Google Scholar 

  37. N. K. Nikol'skii, “Invariant subspaces of weighted shift operators,” Matem. Sb.,74, No. 2, 171–190 (1967).

    Google Scholar 

  38. N. K. Nikol'skii, “Basicity and unicellularity of weighted shift operators,” Funkts. Anal. i Ego Prilozh.,2, No. 2, 95–96 (1968).

    Google Scholar 

  39. N. K. Nikol'skii, “Nonstandard ideals, unicellularity, and algebras related to the shift operator,” Zap. Nauch. Semin. Leningr. Otd. Matem. Inst. Akad. Nauk SSSR,19, 156–195 (1970).

    Google Scholar 

  40. L. S. Pontryagin, “Hermitian operators in a space with indefinite metric,” Izv. Akad. Nauk SSSR. Ser. Matem.,8, No. 1, 243–280 (1944).

    Google Scholar 

  41. L. A. Sakhnovich, “Reduction of Volterra operators to simplest form and inverse problems,” Izv. Akad. Nauk SSSR,21, 235–262 (1957).

    Google Scholar 

  42. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Space [Russian translation], Mir, Moscow (1970), 431 pp.

    Google Scholar 

  43. R. R. Phelps, Lectures on Choquet's Theorem [Russian translation], Mir, Moscow (1968).

    Google Scholar 

  44. P. Halmos, Hilbert Space in Problems [Russian translation], Mir, Moscow (1970), 352 pp.

    Google Scholar 

  45. P. Halmos, “Then problems in the theory of Hilbert spaces,” Matematika,15, No. 4, 28–67 (1971).

    Google Scholar 

  46. P. Halmos, “Multiplication operators in C* -algebras and reflexivity problem,” Dokl Akad. Nauk SSSR,210, No. 3, 543–544 (1973).

    Google Scholar 

  47. P. Halmos, “Reflexive operator algebras,” Matem. Sb.,87, No. 2, 92–93 (1972).

    Google Scholar 

  48. P. Halmos, “Multiplication operators in C* -algebras in reflexivity problems for algebras containing m.a.s.a.,” Funkts. Anal. i Ego Prilozh.,8, No. 1, 92–93 (1974).

    Google Scholar 

  49. P. Halmos, “Operator algebras with strictly cyclic vectors,” Matem. Zametki,16, No. 2, 253–257 (1974).

    Google Scholar 

  50. P. Halmos, “Operator algebras in a typeII1 space with indefinite metric,” Dokl. Akad. Nauk SSSR,201, No. 1, 44–47 (1971).

    Google Scholar 

  51. P. Halmos, “Symmetric Banach operator algebras in a type II1 space,” Matem. Sb.,89, No. 2, 264–279 (1972).

    Google Scholar 

  52. T. Andô, “On a pair of commutative contractions,” Acta Sci. Math.,24, Nos. 1–2, 88–90 (1963).

    Google Scholar 

  53. T. Andô, “A note on invariant subspaces of a compact normal operator,” Arch. Math.,14, Nos. 4–5, 337–340 (1963).

    Google Scholar 

  54. C. Apostol, “Hypercommutativity and invariant subspaces,” Rev. Roum. Math. Pures Appl.,17, No. 3, 335–339 (1972).

    Google Scholar 

  55. C. Apoltol, “Quasitriangularity in Hilbert space,” Indiana Univ. Math. J.,22, No. 9, 817–825 (1973).

    Google Scholar 

  56. C. Apoltol, C. Foias, and L. Zsido, “On non-quasitriangular operators,” C. R. Acad. Sci.,275, No. 10, A501-A503 (1972).

    Google Scholar 

  57. C. Apostol, C. Foias, and D. Voiculescu, “Spectral structure of nonquasitriangular operators,” C. R. Acad. Sci.,276, No. 1, A49-A51 (1973).

    Google Scholar 

  58. C. Apostol, C. Foias, and D. Voisculescu, “Some results on non-quasitriangular operators. II,” Rev. Roum. Math. Pures Appl.,18, No. 2, 159–181 (1973).

    Google Scholar 

  59. C. Apostol, C. Foias, and D. Voisculescu, “Some results on nonquasitriangular operators. III,” Rev. Roumn. Math. Pures Appl.,18, No. 3, 309–324 (1973).

    Google Scholar 

  60. C. Apostol, C. Foias, and D. Voisculescu, “Some results on nonquasitriangular operators. IV,” Rev. Roumn. Math. Pures Appl.,18, No. 4, 487–514 (1973).

    Google Scholar 

  61. W. B. Arveson, “A density theorem for operator algebras,” Duke Math. J.,34, 635–647 (1967).

    Google Scholar 

  62. W. B. Arveson, “An algebraic conjugacy invariant for measure preserving automorphisms,” Bull. Amer. Math. Soc.,73, No. 1, 121–125 (1967).

    Google Scholar 

  63. W. B. Arveson, “Operator algebras and measure preserving automorphisms,” Acta Math.,118, Nos. 1–2, 95–109 (1967).

    Google Scholar 

  64. W. B. Arveson, “Analyticity in operator algebras,” Amer. J. Math.,89, No. 3, 578–642 (1967).

    Google Scholar 

  65. W. B. Arveson, “On subalgebras of C*-algebras,” Bull. Amer. Math. Soc.,75, No. 4, 790–794 (1969).

    Google Scholar 

  66. W. B. Arveson, “Subalgebras of C*-algebras. I,” Acta Math.,123, Nos. 3–4, 141–224 (1969).

    Google Scholar 

  67. W. B. Arveson, “Subalgebras of C*-algebras. II,” Acta Math.,128, Nos. 3–4, 271–308 (1972).

    Google Scholar 

  68. W. B. Arveson, “Unitary invariants for compact operators,” Bull. Amer. Math. Soc.,76, No. 1, 88–91 (1970).

    Google Scholar 

  69. W. B. Arveson, “Lattices of invariant subspaces,” Bull. Amer. Math. Soc.,78, No. 4, 515–519 (1972).

    Google Scholar 

  70. W. B. Arveson and J. Feldman, “A note on invariant subspaces,” Mich. Math. J.,15, No. 1, 61–64 (1968).

    Google Scholar 

  71. W. B. Arveson and K. B. Josephson, “Operator algebras and measure preserving automorphisms. II,” J. Funct. Anal.,4, No. 1, 100–134 (1969).

    Google Scholar 

  72. W. G. Bade, “Weak and strong limits of spectral operators,” Pacif. J. Math.,4, No. 3, 393–413 (1954).

    Google Scholar 

  73. B. A. Barnes, “Density theorem for algebras of operators and annihilator Banach algebras,” Mich. Math. J.,19, No. 2, 149–155 (1972).

    Google Scholar 

  74. B. A. Barnes, “Irreducible algebras of operators which contain a minimal idempotent,” Proc. Amer. Math. Soc.,30, No. 2, 337–342 (1971).

    Google Scholar 

  75. H. Behncke, “Structure of certain nonnormal operators. II,” Indiana Univ. Math. J.,22, No. 4, 301–308 (1972).

    Google Scholar 

  76. C. A. Berger, “Normal dilations,” Doctoral Dissertation, Cornell Univ. (1963), 67 pp.; Dissert. Abstrs.,24, No. 7, 2918 (1964).

  77. A. Bernstein and A. Robinson, “Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos,” Pacif. J. Math.,16, No. 3, 421–431 (1966).

    Google Scholar 

  78. A. Beurling, “On two problems concerning linear transformations in Hilbert space,” Acta Math.,81, 239–255 (1949).

    Google Scholar 

  79. E. Bishop, “A generalization of the Stone-Weierstrass theorem,” Pacif. J. Math.,11, No. 3, 777–783 (1961).

    Google Scholar 

  80. R. Bolstein, “Strictly cyclic operators,” Duke Math. J.,40, No. 3, 683–688 (1973).

    Google Scholar 

  81. R. Bolstein and W. Wogen, “Subnormal operators in strictly cyclic operator algebras,” Pacif. J. Math.,49, No. 1, 7–11 (1973).

    Google Scholar 

  82. R. Bouldin, “Reducing decomposition for strictly cyclic operators,” Proc. Amer. Math. Soc.,40, No. 2, 477–481 (1973).

    Google Scholar 

  83. J. Bram, “Subnormal operators,” Duke Math. J.,22, No. 1, 75–94 (1955).

    Google Scholar 

  84. S. Brehmer, “Uber vertauschbare Kontractionen des Hilbertschen Raumes,” Acta Sci. Math.,22, Nos. 1–2, 106–111 (1961).

    Google Scholar 

  85. J. E. Brennan, “Point evaluation and invariant subspaces,” Indiana Univ. Math. J.,20, No. 10, 879–881 (1971).

    Google Scholar 

  86. J. E. Brennan, “Invariant subspaces and rational approximation,” J. Funct. Anal.,7, No. 2, 285–310 (1971).

    Google Scholar 

  87. L. Brickman and P. A. Fillmore, “The invariant subspace lattice of a linear transformation,” Can. J. Math.,19, No. 4, 810–822 (1967).

    Google Scholar 

  88. L. Brown, A. Shields, and K. Zeller, “On absolutely convergent exponential sums,” Trans. Amer. Math. Soc.,96, No. 1, 162–183 (1960).

    Google Scholar 

  89. Doan Khanh Bui, “Invariant subspaces of normal operators,” Bull. Sci. Math.,93, Nos. 3–4, 175–180 (1969–1970).

    Google Scholar 

  90. J. W. Bunce and J. A. Deddens, “Irreducible representation of the C*-algebras generated by an n-normal operator,” Trans. Amer. Math. Soc.,171, Sept., 301–307 (1973).

    Google Scholar 

  91. J. W. Bunce and J. A. Deddens, “C*-algebras generated by weighted shifts,” Indiana Univ. Math. J.,23, No. 3, 257–271 (1973).

    Google Scholar 

  92. M.-D. Choi, “Positive linear maps on C*-algebras,” Can. J. Math.,24, No. 3, 520–529 (1972).

    Google Scholar 

  93. L. A. Coburn, “The C*-aigebra generated by an isometry,” Bull. Amer. Math. Soc.,73, No. 5, 722–726 (1967).

    Google Scholar 

  94. L. A. Coburn, “The C*-algebra generated by an isometry. II,” Trans. Amer. Math. Soc.,137, 211–217 (1969).

    Google Scholar 

  95. J. B. Conway, “On algebras of operators with totally ordered lattices of invariant subspaces,” Proc. Amer. Math. Soc.,28, No. 1, 163–169 (1971).

    Google Scholar 

  96. T. Crimmins and P. Rosenthal, “On decomposition of invariant subspacss,” Bull. Amer. Math. Soc.,73, No. 1 (1967).

  97. C. Davis, H. Radjavl and P. Rosenthal, “On operator algebras and invariant subspaces,” Can. J. Math.,21, No. 5, 1178–1181 (1969).

    Google Scholar 

  98. D. Deckard, R. G. Douglas and C. Pearcy, “On invariant subspaces of quasi-triangular operators,” Amer. J. Math.,51, No. 3, 637–647 (1969).

    Google Scholar 

  99. J. A. Deddens, “Reflexive operators,” Indiana Univ. Math. J.,20, No. 10, 887–889 (1971).

    Google Scholar 

  100. J. A. Deddens, “Every isometry is reflexive,” Proc. Amer. Math. Soc.,28, No. 2, 509–512 (1971).

    Google Scholar 

  101. J. A. Deddens, “Intertwining analytic Toeplitz operators,” Mich. Math. J.,18, No. 3, 243–246 (1971).

    Google Scholar 

  102. J. Dixmier, Operator Algebras in Hilbert Spaces (von Neumann Algebras), 2nd ed. rev. and suppl., Gauthier-Villars, Paris (1969), 367 pp.

    Google Scholar 

  103. W. F. Donoghue, Jr., “The lattice of invariant of a completely continuous quasi-nilpotent transformation,” Pacif. J. Math.,7, No. 2, 1031–1035 (1957).

    Google Scholar 

  104. R. G. Douglas, “On the hyperinvariant subspaces of isometries,” Math. Z.,107, No. 4, 297–300 (1968).

    Google Scholar 

  105. R. G. Douglas, “On the operator equations S*XT=X and related topics,” Acta Sci. Math.,30, Nos. 1–2, 19–32 (1969).

    Google Scholar 

  106. R. G. Douglas, “On the C*-algebras of a one-parameter semigroup of isometries,” Acta Math.,128, Nos. 3–4, 143–151 (1972).

    Google Scholar 

  107. R. G. Douglas and C. Pearcy, “On topology for invariant subspaces,” J. Funct. Anal.,2, No. 3, 323–341 (1968).

    Google Scholar 

  108. R. G. Douglas and C. Pearcy, “A note on quasitriangular operators,” Duke Math. J.,37, No. 1, 177–188 (1970).

    Google Scholar 

  109. R. G. Douglas and C. Pearcy, “Hyperinvariant subspaces and transitive algebras,” Mich. Math., J.,19, No. 1, 1–12 (1972).

    Google Scholar 

  110. H. A. Dye, “On groups of measure-preserving transformation. I,” Amer. J. Math.,81, No. 1, 119–159 (1959).

    Google Scholar 

  111. J. A. Dyer, E. A. Pedersen, and P. Porcelli, “An equivalent formulation of the invariant subspace conjecture,” Bull. Amer. Math. Soc.,78, No. 6, 1020–1023 (1972).

    Google Scholar 

  112. M. R. Embry, “On invariant subspace theorem,” Proc. Amer. Math. Soc.,32, No. 1, 331–332 (1972).

    Google Scholar 

  113. M. R. Embry, “Strictly cyclic operator algebras on a Banach space,” Pacif. J. Math.,45, No. 2, 443–452 (1973).

    Google Scholar 

  114. J. A. Erdös, “Some results on triangular operator algebras,” Amer. J. Math.,89, No. 1, 85–93 (1967).

    Google Scholar 

  115. J. M. G. Fell, “The structure of algebras of operator fields,” Acta Math.,106, Nos. 3–4, 233–280 (1961).

    Google Scholar 

  116. C. Foias, “Unele aplicatii ale multimilor spectrale. I. Masura armonică-spectrală. Studii si cerce-tari.,” Mat. Acad. RPR,10, No. 2, 365–401 (1959).

    Google Scholar 

  117. C. Foias, “Spectral maximal spaces and decomposable operators in Banach space,” Arch. Math.,14, Nos. 4–5, 341–349 (1963).

    Google Scholar 

  118. C. Foias, “Invariant para-closed subspaces,” Indiana Univ. Math. J.,20, No. 10, 897–900 (1971).

    Google Scholar 

  119. C. Foias, “Invariant para-closed subspaces,” Indiana Univ. Math. J.,21, No. 10, 887–906 (1972).

    Google Scholar 

  120. C. Foias, “On the scalar parts of a decomposable operator,” Rev. Roumn. Math. Pures Appl.17, No. 8, 1181–1198 (1972).

    Google Scholar 

  121. C. Foias and J. P. Williams, “Some remarks on Volterra operators,” Proc. Amer. Math. Soc.,31, No. 1, 177–184 (1972).

    Google Scholar 

  122. B. Fuglede, “A commutativity theorem for normal operators,” Proc. Amer. Mali. Soc.,36, 35–40 (1950).

    Google Scholar 

  123. B. Fuglede and R. Kadison, “Determinant theory in finite factors,” Ann. Math.55, No. 3, 520–530 (1952).

    Google Scholar 

  124. F. Gilfeather, “On the Suzuki structure theory fornonself-adjoint operators on Hilbert space,” Acta Sci. Math.,32, Nos. 3–4, 239–250 (1971).

    Google Scholar 

  125. J. Glimm, “A Stone-Weierstrass theorem for C*-algebras,” Ann. Math.,72, No. 2, 216–244 (1960).

    Google Scholar 

  126. R. Goodman, “Invariant subspaces for normal operators,” J. Math. Mech.,15, No. 1, 123–128 (1966).

    Google Scholar 

  127. P. R. Halmos, “Invariant subspaces for polynomially compact operators,” Pacif. J. Math.,16, No. 3, 433–437 (1966).

    Google Scholar 

  128. P. R. Halmos, “Quasitriangular operators,” Acta Sci. Math.,29, Nos. 3–4, 283–294 (1968).

    Google Scholar 

  129. P. R. Halmos, “Invariant subspaces. Abstract spaces and approximation,” Proc. M. R. I. Oberwolfach, Birkhäuser, Basel (1968), pp. 26–30.

    Google Scholar 

  130. P. R. Halmos, “Capacity in Banach algebras,” Indiana Univ. Math. J.,20, No. 9, 855–863 (1971).

    Google Scholar 

  131. K. J. Harrison, “Transitive atomic lattices of subspaces,” Indiana Univ. Math. J.,21, No. 7, 621–642 (1972).

    Google Scholar 

  132. K. J. Harrison, H. Radjavi, and P. Rosenthal, “A transitive medial subspace lattice,” Proc. Amer. Math. Soc.,28, No. 1, 119–121 (1971).

    Google Scholar 

  133. H. Helson and D. Lowdenslager, “Prediction theory and Fourier series in several variables,” Acta Math.,99, Nos. 3–4, 165–202 (1958).

    Google Scholar 

  134. D. A. Herrero, “A pathological lattice of invariant subspaces,” J. Funct. Anal.,11, No. 2, 131–137 (1972).

    Google Scholar 

  135. D. A. Herrero, “Operator algebras of finite strict multiplicity,” Indiana Univ. Math. J.,22, No. 1, 13–24 (1972).

    Google Scholar 

  136. D. A. Herrero, “Algebras de operadores transitivas que contienen una subalgebra de multiplicidad estricta finita,” Rev. Union Mat. Argent.,26, No. 2, 77–84 (1972).

    Google Scholar 

  137. D. A. Herrero and N. Salinas, “Analytically-invariant and bi-invariant subspaces,” Trans. Amer. Math. Soc.,173, Nov., 117–136 (1972).

    Google Scholar 

  138. R. A. Hirshfeld, “On polynomials in several Hilbert space operators,” Math. Z.,127, No. 3, 224–234 (1972).

    Google Scholar 

  139. J. A. R. Holbrook, “Spectral dilations and polynomially bounded operators,” Indiana Univ. Math. J.,20, No. 11, 1027–1034 (1971).

    Google Scholar 

  140. T. B. Hoover, “Hyperinvariant subspaces for n-normal operators,” Acta Sci. Math.,32, Nos. 1–2, 109–119 (1971).

    Google Scholar 

  141. T. B. Hoover, “Operator algebras with complemented invariant subspace lattices,” Indiana Univ. Math. J.,22, No. 11, 1029–1035 (1973).

    Google Scholar 

  142. T. B. Hoover, “Operator algebras with reducing invariant subspaces,” Pacif. J. Math.,44, No. 1, 173–179 (1973).

    Google Scholar 

  143. A. Hopenwasser, “Isometries on irreducible triangular operator algebras,” Math. Scand.,30, No. 1, 136–140 (1972).

    Google Scholar 

  144. A. Hopenwasser, “Completely isometric maps and triangular operator algebras,” Proc. London Math. Soc.,25, No. 1, 96–114 (1972).

    Google Scholar 

  145. T. Itô, “On the commutative family of subnormal operators,” J. Fac. Sci. Hokkaido Univ., Ser. 1,14, 1–15 (1958).

    Google Scholar 

  146. B. E. Johnson and A. L. Shields, “Hyperinvariant subspaces for operators on the space of complex sequences,” Mich. Math. J.,19, No. 2, 189–191 (1972).

    Google Scholar 

  147. R. E. Johnson, “Distinguished rings of linear transformations,” Trans. Amer. Math. Soc.,111, 400–412 (1964).

    Google Scholar 

  148. R. V. Kadison and J. M. Singer, “Triangular operator algebras,” Amer. J. Math.,82, No. 2, 227–259 (1960).

    Google Scholar 

  149. G. Kalisch, “On similarity, reducing manifolds, and unitary equivalence of certain Volterra operators,” Ann. Math.,66, No. 3, 481–494 (1957).

    Google Scholar 

  150. N. Kamei, “Simply invariant subspace theorem for antisymmetric subdiagonal algebras,” Tôhoku Math. J.,21, No. 3, 467–473 (1969).

    Google Scholar 

  151. K.-M. Körber, “Die invarianten Teilräume der stetigen Endomorphismen von ω,” Math. Ann.,182, No. 2, 95–103 (1969).

    Google Scholar 

  152. A. Lambert, “Strictly cyclic weighted shifts,” Proc. Amer. Math. Soc.,29, No. 2, 331–336 (1972).

    Google Scholar 

  153. A. Lambert, “Strictly cyclic operator algebras,” Pacif. J. Math.,39, No. 3, 717–726 (1971).

    Google Scholar 

  154. A. Lambert, “Spectral properties of strictly cyclic operator algebras,” Indiana Univ. Math. J.,22, No. 10, 959–963 (1973).

    Google Scholar 

  155. A. Lambert, “The algebra generated by an invertibly weighted shift,” J. London Math. Soc.,5, No. 4, 741–747 (1972).

    Google Scholar 

  156. E. C. Lance, “Some properties of nest algebras,” Proc. London Math. Soc.,19, No. 1, 45–68 (1969).

    Google Scholar 

  157. P. D. Lax, “Translation invariant subspaces,” Acta Math.,101, Nos. 3–4, 163–178 (1959).

    Google Scholar 

  158. A. Lebow, “On von Neumann's theory of spectral sets,” J. Math. Anal. and Appl.,7, No. 1, 64–90 (1963).

    Google Scholar 

  159. R. Leggett, “On the invariant subspace structure of compact dissipative operators,” Indiana Univ. Math. J.,22, No. 10, 919–928 (1973).

    Google Scholar 

  160. G. Lumer, “States, quotient algebras and invariant subspaces,” C. R. Acad. Sci.,274, No. 17, A1308-A1311 (1972).

    Google Scholar 

  161. G. Lumer and M. Rosenblum, “Linear operator equations,” Proc. Amer. Math. Soc.,10, No. 1, 32–41 (1959).

    Google Scholar 

  162. M. A. Naimark, “On commuting unitary operators in spaces with indefinite metric,” Acta Sci. Math.,24, Nos. 3–4, 177–189 (1963).

    Google Scholar 

  163. M. A. Naimark, “Kommutative symmetrische Operatorenalgebren in Pontrjyaginschen Raumen IIk,” Math. Ann.,162, No. 1, 147–171 (1965).

    Google Scholar 

  164. J. von Neumann, “On rings of operators, III,” Ann. Math.,41, 94–161 (1940).

    Google Scholar 

  165. J. von Neumann, “Eine Spectralthoerie für allgemeine Operatoren eines unitären Raumes,” Math. Nach.,4, 258–281 (1951).

    Google Scholar 

  166. E. A. Nordgren, “Invariant subspaces of a direct sum of weighted shifts,” Pacif. J. Math.,27, No. 3, 587–598 (1968).

    Google Scholar 

  167. E. A. Nordgren, “Transitive algebras,” Indiana Univ. Math. J.,20, No. 10, 925–927 (1971).

    Google Scholar 

  168. E. A. Nordgren, “Transitive operator algebras,” J. Math. Anal. Appl.,32, No. 3, 639–643 (1970).

    Google Scholar 

  169. E. A. Nordgren, H. Radjavi, and P. Rosenthal, “On density of transitive algebras,” Acta. Sci. Math.,3, Nos. 3–4 (1969).

  170. E. A. Nordgren and P. Rosenthal, “Algebras containing unilateral shifts or finite-rank operators,” Duke Math. J.,40, No. 2, 419–424 (1973).

    Google Scholar 

  171. S. Parrot, “Unitary dilations for commuting contractions,” Pacif. J. Math.,34, No. 2, 481–490 (1970).

    Google Scholar 

  172. C. Pearcy, “On certain von Neumann algebras which are generated by partial isometries,” Proc. Amer. Soc.,15, No. 3, 393–395 (1964).

    Google Scholar 

  173. C. Pearcy and N. Salinas, “An invariant-subspace theorem,” Mich. Math. J.,20, No. 1, 23–31 (1973).

    Google Scholar 

  174. F. M. Pollack, “Properties of the matrix range of an operator,” Indiana Univ. Math. J.,22, No. 5, 419–427 (1972).

    Google Scholar 

  175. H. Radjavi and P. Rosenthal, “Invariant subspaces and weakly closed algebras,” Bull. Amer. Math. Soc.,74, No. 5, 1013–1014 (1968).

    Google Scholar 

  176. H. Radjavi and P. Rosenthal, “On invariant subspaces and reflexive algebras,” Amer. J. Math.,51, No. 1, 683–692 (1969).

    Google Scholar 

  177. H. Radjavi and P. Rosenthal, “On reflexive algebras of operators,” Indiana Univ. Math. J.,20, No. 10, 935–937 (1971).

    Google Scholar 

  178. H. Radjavi and P. Rosenthal, “A sufficient condition that an operator algebra be self-adjoint,” Can. J. Math.,23, No. 4, 588–597 (1971).

    Google Scholar 

  179. H. Radjavi and P. Rosenthal, “Hyperinvariant subspaces for spectral and n-normal operators,” Acta Sci. Math.,32, Nos. 1–2, 121–126 (1971).

    Google Scholar 

  180. C. E. Rickart, “The uniqueness of norm problem in Banach algebras,” Ann. Math.,51, 615–628 (1950).

    Google Scholar 

  181. J. R. Ringrose, “On some algebras of operators,” Proc. London Math. Soc.,15, No. 3, 61–83 (1965).

    Google Scholar 

  182. J. R. Ringrose, “Algebraic isomorphisms between ordered bases,” Amer. J. Math.,83, No. 3, 463–478 (1961).

    Google Scholar 

  183. J. R. Ringrose, “On some algebras of operators, II,” Proc. London Math. Soc.,16, No. 3, 385–402 (1966).

    Google Scholar 

  184. P. Rosenthal, “A note on unicellular operators,” Proc. Amer. Math. Soc.,19, No. 2, 505–506 (1968).

    Google Scholar 

  185. P. Rosenthal, “Completely reducible operators,” Proc. Amer. Math. Soc.,19, No. 4, 826–830 (1968).

    Google Scholar 

  186. P. Rosenthal, “Examples of invariant subspace lattices,” Duke Math. J.,37, No. 1, 103–112 (1970).

    Google Scholar 

  187. P. Rosenthal, “Remarks on invariant subspace lattices,” Can. Math. Bull.,12, No. 5, 639–643 (1969).

    Google Scholar 

  188. P. Rosenthal, “Weakly closed maximal triangular algebras are hyperreducible,” Proc. Amer. Math. Soc.,24, No. 1, 220 (1970).

    Google Scholar 

  189. T. Saitô, “Some remarks to Andô's theorem,” Tôhoku Math. J.,18, No. 4, 404–409 (1966).

    Google Scholar 

  190. D. E. Sarason, “A remark on the Volterra operators,” J. Math. Anal. Appl.,12, No. 2, 244–246 (1965).

    Google Scholar 

  191. D. E. Sarason, “Invariant subspaces and unstarred operator algebras,” Pacif. J. Math.,17, No. 3, 511–517 (1966).

    Google Scholar 

  192. D. E. Sarason, “Weak-star density of polynomials,” J. Reine Angew. Math.,252, 1–15 (1972).

    Google Scholar 

  193. H. Schaefer, “Eine Bemerkung zur Existenz invaruanter Tailräume linearer Abbildungen,” Math. Z.,82, No. 1, 90 (1963).

    Google Scholar 

  194. M. Schreiber, “A functional calculus for general operators in Hilbert space,” Trans. Amer. Math. Soc.,87, No. 1, 108–118 (1958).

    Google Scholar 

  195. J. T. Schwartz, “On spectral operators in Hilbert space with compact imaginary part,” Commun. Pure Appl. Math.,15, No. 1, 95–97 (1962).

    Google Scholar 

  196. J. T. Schwartz, “Subdiagonalization of operators in Hilbert space with compact imaginary part,” Commun. Pure Appl. Math.,15, No. 2, 159–172 (1962).

    Google Scholar 

  197. J. T. Scroggs, “Invariant subspaces of a normal operator,” Duke Math. J.,26, No. 1, 85–112 (1959).

    Google Scholar 

  198. I. Segal, “A noncommutative extension of abstract integration,” Ann. Math.,57, No. 3, 401–457 (1953).

    Google Scholar 

  199. A. L. Shields, “A note on invariant subspaces,” Mich. Math. J.,17, No. 3, 231–233 (1970).

    Google Scholar 

  200. T. Srinivasan and J. Wang, “Weak* Dirichlet algebras,” Symposium on Function Algebras, Scott-Foresman (1966).

  201. W. Stinespring, “Positive functions on C*-algebras,” Proc. Amer. Math. Soc.,6, No. 2, 211–216 (1955).

    Google Scholar 

  202. N. Suzuki, “Algebraic aspects of non-self-adjoint operators,” Proc. Japan. Acad.,41, No. 8, 706–710 (1965).

    Google Scholar 

  203. N. Suzuki, “The algebraic structure of non-self-adjoint operators,” Acta Sci. Math.,27, Nos. 3–4, 173–184 (1966).

    Google Scholar 

  204. N. Suzuki, “The structure of spectral operators with completely continuous imaginary part,” Proc. Amer. Math. Soc.,22, No. 1, 82–84 (1969).

    Google Scholar 

  205. N. Suzuki, “Reduction theory of operators on Hilbert space. The invariant subspace problem,” Indiana Univ. Math. J.,20, No. 10, 953–958 (1971).

    Google Scholar 

  206. B. Sz.-Nagy, “On uniformly bounded linear operators in Hilbert space,” Acta Sci. Math.,11, 152–157 (1947).

    Google Scholar 

  207. B. Sz.-Nagy “Contractions of Hilbert space,” Acta Sci. Math.,15, No. 1, 87–92 (1953).

    Google Scholar 

  208. B. Sz.-Nagy, “Transformations of Hilbert space and positive-type functions on a group,” Acta Sci. Math.,15, No. 2, 104–114 (1954).

    Google Scholar 

  209. B. Sz.-Nagy and C. Foias, “Quasi-similarity between operators and invariant subspaces,” C. R. Acad. Sci.,261, No. 20, 3938–3940 (1965).

    Google Scholar 

  210. B. Sz.-Nagy and C. Foias, “Contractions of Hilbert space. III,” Acta Sci. Math.,19, Nos. 1–2, 26–46 (1958).

    Google Scholar 

  211. B. Sz.-Nagy and C. Foias, “Contractions of Hilbert space. IV,” Acta Sci. Math.,21, Nos. 3–4, 251–259 (1960).

    Google Scholar 

  212. B. Sz.-Nagy and C. Foias, “Contractions of Hilbert space. VI. Functional calculus,” Acta Sci. Math.,23, Nos. 1–2, 130–167 (1962).

    Google Scholar 

  213. B. Sz.-Nagy and C. Foias, “Contractions of Hilbert space. VII. Canonical triangulation. Minimal functions,” Acta Sci. Math.,25, Nos. 1–2, 12–37 (1964).

    Google Scholar 

  214. B. Sz.-Nagy and C. Foias, “Contractions of Hilbert Space, VIII. Characteristic functions. Functional models,” Acta Sci. Math.,25, Nos. 1–2, 38–72 (1964).

    Google Scholar 

  215. B. Sz.-Nagy and C. Foias, “Operators without multiplicity,” Acta. Sci. Math.,30, Nos. 1–2, 1–18 (1969).

    Google Scholar 

  216. B. Sz.-Nagy and C. Foias, “Jordan model for a class of operators in Hilbert space,” Acta Sci. Math.,31, Nos. 1–2, 91–115 (1970).

    Google Scholar 

  217. B. Sz.-Nagy and C. Foias, “Cyclic and quasi-affine vectors,” Stud. Math.,31, No. 1, 35–42 (1968).

    Google Scholar 

  218. D. K. Taylor, “Interpolation in algebras of operator fields,” J. Funct. Anal.,10, No. 2, 159–190 (1972).

    Google Scholar 

  219. Jun Tomiyama, “On some types of maximal Abelian subalgebras,” J. Funct. Anal.,10, No. 4, 373–386 (1972).

    Google Scholar 

  220. J. Wermer, “On invariant subspaces of normal operators,” Proc. Amer. Math. Soc.,2, 270–277 (1952).

    Google Scholar 

  221. N. Wiener, “On the factorization of matrices,” Comment. Math. Helv.,29, No. 2, 97–111 (1955).

    Google Scholar 

  222. B. Yood, “Additive groups and linear manifolds of transformations between Banach spaces,” Amer. J. Math.,71, 663–677 (1949).

    Google Scholar 

  223. Takashi Yoshino, “Subnormal operator with a cyclic vector,” Tôhoku Math. J.,21, No. 1, 47–55 (1969).

    Google Scholar 

Download references

Authors
  1. M. A. Naimark
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Loginov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. S. Shul'man
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki i Tekhniki. Matamaticheskii Analiz., Part 2, Vol. 10, pp. 413–465, 1974.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Naimark, M.A., Loginov, A.I. & Shul'man, V.S. Non-self-adjoint operator algebras in Hilbert space. J Math Sci 5, 250–278 (1976). https://doi.org/10.1007/BF01247398

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01247398

Keywords

Search

Navigation