Alexey
Popov,
Department
of
Pure
Mathematics,
University
of
Waterloo
“On
the
spatial
structure
of
semigroups
of
partial
isometries.”
It
is
a
well-known
fact
that
if
a
group
of
matrices
is
bounded,
then
it
is
simultaneously
similar
to
a
group
of
unitaries.
We
will
start
this
talk
by
discussing
a
semigroup
analogue
of
this
statement
(a
set
of
operators
is
called
a
semigroup
if
it
is
closed
under
multiplication).
Then
we
will
investigate
the
spatial
structure
of
semigroups
of
partial
isometries
on
a
separable
Hilbert
space.
In
the
end
of
the
talk,
we
will
show
that
an
irreducible
(that
is,
having
no
common
invariant
subspaces)
matrix
semigroup
is
similar
to
a
semigroup
of
partial
isometries
if
and
only
if
(a)
the
norms
of
nonzero
members
of
it
are
uniformly
bounded
above
and
below
and
(b)
its
idempotents
commute.