Preprint: Frames in Hilbert C* modules and algebras

[Michael Frank (frank@math.uh.edu)]

#8 Preprint: Frames in Hilbert C* modules and algebras

Dear colleagues,

I would like to announce the existence of two preprints that are related
to questions in wavelet-theory. The authors would be glad if the
information could be included into the wavelet newsletter.

Thank you very much for your support,
greatings and best wishes

Michael Frank.

TITLE:
Frames in Hilbert C*-modules and C*-algebras

AUTHORS:
Michael Frank
Dept. Mathematics, Univ. of Houston, Houston, TX 77204-3476, U.S.A.
frank@math.uh.edu, frank@mathematik.uni-leipzig.de
David R. Larson
Dept. Mathematics, Texas A&M Univ., College Station, TX 77843, U.S.A.
larson@math.tamu.edu

HOWPUBLISHED:
preprint, LaTeX2e, 33 pages, submitted. Will be presented at the Joint
Mathematics Meeting in San Antonio, Texas, U.S.A., on January 13th, 1998.
(Section: The Functional and Harmonic Analysis of Wavelets.)

ABSTRACT:
We present a general approach to a module frame theory in C*-algebras and
Hilbert C*-modules. The investigations rely on the idea of geometric dilation
to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal
Hilbert bases, and of reconstruction of the frames by projections and other
bounded module operators with suitable ranges. We obtain frame representation
and decomposition theorems, as well as similarity and equivalence results.
The Hilbert space situation appears as a special case. Using a canonical
categorical equivalence of Hilbert C*-modules over commutative C*-algebras
and (F)Hilbert bundles the results find a reinterpretation for frames in
vector and (F)Hilbert bundles.
The results are of interest in wavelet theory (esp. MRA wavelets), Hilbert
C*-module and C*-algebra theory, vector bundle theory. Particular results
on Hilbert-Schmidt operators are even new in the Hilbert space situation.
We show that any algebraically generating set of an algebraically finitely
generated Hilbert C*-module is a module frame.

TITLE:
Symmetric approximation of frames and bases in Hilbert spaces

AUTHORS:
Michael Frank
Dept. Mathematics, Univ. of Houston, Houston, TX 77204-3476, U.S.A.
frank@math.uh.edu, frank@mathematik.uni-leipzig.de
Vern I. Paulsen
Dept. Mathematics, Univ. of Houston, Houston, TX 77204-3476, U.S.A.
vern@math.uh.edu
Terry R. Tiballi
Dept. Mathematics, SUNY at Oswego, Oswego, NY 13126, U.S.A.
tiballi@oswego.edu

HOWPUBLISHED:
preprint, LaTeX2e, 16 pages, submitted.

ABSTRACT:
We consider existence and uniqueness of symmetric approximation of frames
by normalized tight frames and of symmetric orthogonalization of bases by
orthonormal bases in Hilbert spaces H . More precisely, we determine whether
a given frame or basis possesses a normalized tight frame or orthonormal
basis that is quadratically closest to it, if there exists such frames
or bases at all. A crucial role is played by the Hilbert-Schmidt property
of the operator (P-|F|) , where F is the adjoint operator of the frame
transform F*: H --> l_2 of the initial frame or basis and (1-P) is the
projection onto the kernel of F .