AG Lie-Gruppen: M. Preeg, FAU: Multiplication Operators in the Unitary Restricted Operators

Multiplication Operators
in the Unitary Restricted Operators – Vortragender: Michael Preeg, Friedrich-Alexander-Universität Erlangen-Nürnberg – Einladender: K.-H. Neeb

Abstract: In
accordance with a projection P on a Hilbert space H, such that H = PH + (1-P)H,
an irreducible representation of the CAR algebra CAR(H) can be constructed on
the tensor product Gamma_P(H) of the Fock spaces F(PH) and the complex
conjugate of F((1-P)H). Unitary operators U on H for which [U,P] is a
Hilbert-Schmidt operator form the restricted unitary operators group
U_(res)(H). For such a restricted operator, the Bogoliubov transformation
Bog(u) of CAR(H) is implementable. This talk examines unitary one-parameter
subgroups of restricted operators, with a particular focus on multiplication
operators of bounded functions on square-integrable functions with the unit
circle as the domain in the case of projection onto the space H_+ of positive Fourier
modes. This leads to the bounded functions of the Sobolev space of fractional
order 1/2. In general, restricted operators can be expressed as an
operator-valued 2×2 matrix with respect to the splitting of H into PH and
(1-P)H. Their norm is defined by the operator norm on the diagonal entries and
the Hilbert-Schmidt norm on the off-diagonal entries. This norm enables us to
conclude that the unitary restricted multiplication operators constitute a
Banach-Lie group. To investigate the generators of one-parameter subgroups, we
draw upon tools from infinite-dimensional Lie theory, demonstrating that their
generators are also restricted and exhibiting some properties of the exponential
function. The appropriate topology on these unitary restricted operators is the
so-called P-strong topology, which replaces the operator norm topology on the
diagonal entries with the strong operator topology. In the case of general
restricted operators, we present an examples of a unbounded non-restricted
generator and a characterisation of (possibly unbounded) restricted generators.
In the case of multiplication operators, we demonstrate that every (possibly
unbounded) real-valued function of the Sobolev space of fractional order 1/2
generates a P-strong continuous unitary one-parameter subgroup.