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On intertwining operators

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Abstract

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BεB(H), defineC (A, B) andR (A, B):B(H)→B(H) byC (A, B) X=AX−XB andR(A, B) X=AXB−X. Our purpose in this note is a twofold one. we show firstly that ifA andB *εB (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC n (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A *,B *)X=0. Secondly, it is shown that ifA andB * are contractions withC 0 completely non-unitary parts, thenR n (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A *,B *)X=C (A, B *)X=C (A *,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.

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References

  1. Douglas, R. G.: On the operator equationS * XT=X and related topics. Acta Sci. Math. (Szeged)30, 19–32 (1969).

    Google Scholar 

  2. Duggal, B. P.: Contractions with a unitary part. J. Lond. Math. Soc.31, 131–136 (1985)

    Google Scholar 

  3. Duggal, B. P.: On dominant operators. Arch. Math.46, 353–359 (1986).

    Google Scholar 

  4. Duggal, B. P.: Intertwining contractions: operator equationsAXB-X andAX=XB. In: Proc. Int. Conf. on ‘Invariant subspaces and allied topics’. Univ. of Delhi (India), Dec. 1986. (To appear.)

  5. Goya, E., Saito, T.: On intertwining by an operator having a dense range. Tohôku Math. J.33, 27–31 (1981).

    Google Scholar 

  6. Sz.-Nagy, B., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. Amsterdam: North-Holland. 1970.

    Google Scholar 

  7. Okubo, R.: The unitary part of paranormal operators. Hokkaido Math. J.6, 273–275 (1977).

    Google Scholar 

  8. Patel, J. M., Sheth, I. H.: On intertwining by an operator having a dense range. Indian J. Pure appl. Math.14, 1077–1082 (1983).

    Google Scholar 

  9. Radjabalipour, M.: An extension of Putnam—Fuglede theorem for hyponormal operators. Math. Z.194, 117–120 (1987).

    Google Scholar 

  10. Stampfli, J. G., Wadhwa, B. L.: An asymmetric Putnam—Fuglede theorem for dominant operators. Indiana Univ. Math. J.25, 359–365 (1976).

    Google Scholar 

  11. Stampfli, J. G., Wadhwa, B. L.: On dominant operators. Mh. Math.84, 143–153 (1977).

    Google Scholar 

  12. Takahashi, K.:C 1 contractions with Hilbert—Schmidt defect operators. J. Operator Th.12, 331–347 (1984).

    Google Scholar 

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  1. School of Mathematical Sciences, University of Khartoum, P.O. Box 321, Khartoum, Sudan

    B. P. Duggal

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  1. B. P. Duggal
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Duggal, B.P. On intertwining operators. Monatshefte für Mathematik 106, 139–148 (1988). https://doi.org/10.1007/BF01298834

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  • DOI: https://doi.org/10.1007/BF01298834

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