Main Article Content
The UMD Constants of the Summation Operators
Abstract
The UMD
property of a Banach space is one of the most useful
properties when one thinks about possible applications. This is in particular
due to the boundedness of the vector-valued Hilbert
transform for functions with values in such a space.
Looking at operators
instead of at spaces, it is easy to check that the summation operator does not
have the UMD property. The actual asyn1ptotic behavior
however of the UMD constants computed with martingales of length n
is unknown.
We explain, why it
would be important to know this behavior, rephrase the problem of finding
these UMD constants and give some evidence of how they behave asymptotically.
Mathematics Subject Classification (2000): Primary 46B07; Secondary 46B03, 46B09, 47B10.
Key words: UMD,
martingales, summation operator, superreflexivity,
Hilbert transform.
Quaestiones Mathematicae 27(2004), 111-136.
property of a Banach space is one of the most useful
properties when one thinks about possible applications. This is in particular
due to the boundedness of the vector-valued Hilbert
transform for functions with values in such a space.
Looking at operators
instead of at spaces, it is easy to check that the summation operator does not
have the UMD property. The actual asyn1ptotic behavior
however of the UMD constants computed with martingales of length n
is unknown.
We explain, why it
would be important to know this behavior, rephrase the problem of finding
these UMD constants and give some evidence of how they behave asymptotically.
Mathematics Subject Classification (2000): Primary 46B07; Secondary 46B03, 46B09, 47B10.
Key words: UMD,
martingales, summation operator, superreflexivity,
Hilbert transform.
Quaestiones Mathematicae 27(2004), 111-136.
