Lyudmila Turowska, Chalmers University of technology & Gothenburg University
"Compact
operator
synthesis
and
essential
spectral
synthesis
in
harmonic
analysis"
W.
Arveson
in
his
fundamental
paper
(Ann.
Math
1974)
discovered
an
interplay
between
invariant
subspaces
and
operator
algebras
theory
and
spectral
synthesis
in
harmonic
analysis.
The
notion
of
operator
synthesis
was
proposed.
It
provided
a
powerful
tool
to
study
different
questions
in
harmonic
analysis,
operator
theory,
theory
of
multipliers
and
so
on.
In
this
talk
we
will
discuss
sets
that
are
operator
synthetic
"modulo
compact
operators''
or
"modulo
Schatten
ideals''.
Analogs
of
such
sets
in
harmonic
analysis
are
subsets
$E\subset{\mathbb
T}$
(${\mathbb
T}$
is
the
circle
group)
such
that
any
pseudofunction
supported
in
$E$
(any
psedomeasure
whose
Fourier
transform
is
in
$l^p$
and
supported
in
$E$)
annihilates
functions
in
$A({\mathbb
T})$
vanishing
on
$E$.
We
shall
discuss
different
examples
of
such
sets,
an
analog
of
Malliavin's
theorem
and
applications
to
operator
equations.