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

V. Mandrekar; P. Richard

Studia Mathematica (1993)

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

Abstract

top
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.

How to cite

top

Mandrekar, 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. [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. [2] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in ℒ p -spaces and their applications, Studia Math. 29 (1968), 275-326. Zbl0183.40501
  3. [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. [4] P. Masani, Quasi-isometric measures and their applications, Bull. Amer. Math. Soc. 76 (1970), 427-528. Zbl0207.44001
  5. [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. [6] G. Pisier, Completely bounded maps between sets of Banach space operators, ibid. 39 (1990), 249-277. Zbl0747.46015
  7. [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. [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. [9] M. Rosenberg, Quasi-isometric dilations of operator-valued measures and Grothendieck's inequality, Pacific J. Math. 103 (1982), 135-161. Zbl0509.46039

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.