Operator Algebra Seminar

Mathematics Department, University of Rome Tor Vergata

Next seminars: 

Abstract: The famous Beurling theorem provides a concrete characterization of closed invariant subspaces for the shift on the Hardy space H^2 in the unit disc, stating that every such space is of the form fH^2 , where f is an inner function. This result can also be interpreted in an operator sense by saying that every closed subspace invariant for the shift is the image of H^2  via an isometry. From this perspective, Beurling’s theorem has been extended by Lax, Halmos, and Rovnyak to shifts of any index, proving that a closed subspace is invariant for a shift if and only if it is the image of the space via a quasi-isometry that commutes with the shift (the so-called Beurling-Lax theorem).


In this talk, I will present a generalization of the “concrete” form of Beurling’s theorem for the shift on the direct finite sum of  H^2. I will show that every closed invariant subspace is given, up to multiplication by an inner function, by the intersection of what we call “determinantal spaces”—which, roughly speaking, are the preimages of shift-invariant subspaces of H^2  via a linear operator commuting with the shift and that are constructed through a determinant of certain matrices with entries given by holomorphic bounded functions. The concreteness of such a structure theorem allows us to prove by rather simple algebraic manipulation, as in the classical Beurling theorem, that the only non-trivial maximal closed shift-invariant subspaces are of codimension one. Using the universality of the (backward) shift in the class of operators with defect less than or equal to the index of the shift, this gives a proof of the following result: every bounded linear operator from a Hilbert space into itself whose defect is finite has a non-trivial closed invariant subspace.

The talk is based on a joint work with Eva Gallardo-Gutierrez

Abstract: Starting from the so-called ETH-Approach to Quantum Mechanics we describe fluorescence and the phenomenon of “quantum jumps” in idealized models of atoms coupled to the quantized electromagnetic field. In a limiting regime where the orbital motion of the atoms is neglected and the velocity of light tends to ∞ we derive explicit non-linear stochastic differential equations describing the effective time

evolution of states of individual atoms. These equations give rise to a measure on state trajectories exhibiting quantum jumps. This measure is a quantum- mechanical analogue of the Wiener measure on Brownian paths encountered in the theory of diffusion.

Joint work with J. Fröhlich and Z. Gang

Upcoming speakers:

List of Seminars (from 2021):


Feb 21, 2024, h 16:00
Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata 

Crossing Symmetry and Endomorphisms of Standard Subspaces
Feb 14, 2024, h 16:00
 Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

2023

 Geometric methods for locally compact quantum groups


Dec 20, 2023, h 16:00
Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

Exact measurement schemes for local observables and the preparation of physical local product states


Dec 12, 2023, h 16:00
Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

Relative entropy for states on the CAR algebra

Nov 29, 2023, h 16:00
Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

2022

2021

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