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

Studia Mathematica (1993)

- Volume: 107, Issue: 2, page 101-113
- ISSN: 0039-3223

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topMandrekar, V., and Richard, P.. "Factorization through Hilbert space and the dilation of L(X,Y)-valued measures." Studia Mathematica 107.2 (1993): 101-113. <http://eudml.org/doc/216023>.

@article{Mandrekar1993,

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.},

author = {Mandrekar, V., Richard, P.},

journal = {Studia Mathematica},

keywords = {spectral dilation of operator-valued measure; Hilbertian operators; factorization; algebraic condition; spectral dilation; -valued measure of finite semivariation; factoring a family of operators through a single Hilbert space},

language = {eng},

number = {2},

pages = {101-113},

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

url = {http://eudml.org/doc/216023},

volume = {107},

year = {1993},

}

TY - JOUR

AU - Mandrekar, V.

AU - Richard, P.

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

JO - Studia Mathematica

PY - 1993

VL - 107

IS - 2

SP - 101

EP - 113

AB - 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.

LA - eng

KW - spectral dilation of operator-valued measure; Hilbertian operators; factorization; algebraic condition; spectral dilation; -valued measure of finite semivariation; factoring a family of operators through a single Hilbert space

UR - http://eudml.org/doc/216023

ER -

## References

top- [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 ${\mathcal{L}}_{p}$-spaces and their applications, Studia Math. 29 (1968), 275-326. Zbl0183.40501
- [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. Zbl0625.60042
- [4] P. Masani, Quasi-isometric measures and their applications, Bull. Amer. Math. Soc. 76 (1970), 427-528. Zbl0207.44001
- [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. Zbl0681.60037
- [6] G. Pisier, Completely bounded maps between sets of Banach space operators, ibid. 39 (1990), 249-277. Zbl0747.46015
- [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. Zbl0791.60025
- [9] M. Rosenberg, Quasi-isometric dilations of operator-valued measures and Grothendieck's inequality, Pacific J. Math. 103 (1982), 135-161. Zbl0509.46039

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