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ArticleOriginal scientific text

Title

Factorization through Hilbert space and the dilation of L(X,Y)-valued measures

Authors 1, 2

Affiliations

  1. Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824, U.S.A.
  2. Department of Science and Mathematics, GMI Engineering and Management Institute, Flint, Michigan 48504, U.S.A.

Abstract

We present a general necessary and sufficient algebraic condition for the spectral dilation of a finitely additive L(X,Y)-valued measure of finite semivariation when X and Y are Banach spaces. Using our condition we derive the main results of Rosenberg, Makagon and Salehi, and Miamee without the assumption that X and/or Y are Hilbert spaces. In addition we relate the dilation problem to the problem of factoring a family of operators through a single Hilbert space.

Keywords

ENG
spectral dilation of operator-valued measure, Hilbertian operators, factorization

Bibliography

  1. S. D. Chatterji, Orthogonally scattered dilation of Hilbert space valued set functions, in: Measure Theory, Oberwolfach 1981, D. Kölzow and D. Maharam-Stone (eds.), Lecture Notes in Math. 945, Springer, New York, 1982, 269-281.
  2. J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in ℒp-spaces and their applications, Studia Math. 29 (1968), 275-326.
  3. A. Makagon and H. Salehi, Spectral dilation of operator-valued measures and its application to infinite-dimensional harmonizable processes, ibid. 85 (1987), 257-297.
  4. P. Masani, Quasi-isometric measures and their applications, Bull. Amer. Math. Soc. 76 (1970), 427-528.
  5. A. G. Miamee, Spectral dilation of L(B,H)-valued measures and its application to stationary dilation for Banach space valued processes, Indiana Univ. Math. J. 38 (1989), 841-860.
  6. G. Pisier, Completely bounded maps between sets of Banach space operators, ibid. 39 (1990), 249-277.
  7. G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., Providence, R.I., 1986.
  8. P. Richard, Harmonizability, V-boundedness, and stationary dilation of Banach-valued processes, in: Probability in Banach Spaces, 8, Proc. Eighth Internat. Conf., R. Dudley, M. Hahn and J. Kuelbs (eds.), Birkhäuser, Boston, 1992, 189-205.
  9. M. Rosenberg, Quasi-isometric dilations of operator-valued measures and Grothendieck's inequality, Pacific J. Math. 103 (1982), 135-161.
Studia Mathematica
1993/107/2
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Pages:
101-113
Main language of publication
English
Received
1991-04-16
Accepted
1993-04-20
Published
1993
Exact and natural sciences