Search results for "Invariant subspace"

Quasi-isometries associated to A-contractions

2014

Abstract Given two operators A and T ( A ≥ 0 , ‖ A ‖ = 1 ) on a Hilbert space H satisfying T ⁎ A T ≤ A , we study the maximum subspace of H which reduces M = A 1 / 2 T to a quasi-isometry, that is on which the equality M ⁎ M = M ⁎ 2 M 2 holds. In some cases, this subspace coincides with the maximum subspace which reduces M to a normal partial isometry, for example when A = T T ⁎ , and in particular if T ⁎ is a cohyponormal contraction. In this case the corresponding subspace can be completely described in terms of asymptotic limit of the contraction T. When M is quasinormal and M ⁎ M = A then the former above quoted subspace reduces to the kernel of A − A 2 . The case of an arbitrary contra…

Operators intertwining with isometries and Brownian parts of 2-isometries

2016

Abstract For two operators A and T ( A ≥ 0 ) on a Hilbert space H satisfying T ⁎ A T = A and the A-regularity condition A T = A 1 / 2 T A 1 / 2 we study the subspace N ( A − A 2 ) in connection with N ( A T − T A ) , for T belonging to different classes. Our results generalize those due to C. Kubrusly concerning the case when T is a contraction and A = S T is the asymptotic limit of T. Also, the particular case of a 2-isometry in the sense of S. Richter as well as J. Agler and M. Stankus is considered. For such operators, under the same regularity condition we completely describe the reducing Brownian unitary and isometric parts, as well as the invariant Brownian isometric part. Some exampl…

Partial isometries and the conjecture of C.K. Fong and S.K. Tsui

2016

Abstract We investigate some bounded linear operators T on a Hilbert space which satisfy the condition | T | ≤ | Re T | . We describe the maximum invariant subspace for a contraction T on which T is a partial isometry to obtain that, in certain cases, the above condition ensures that T is self-adjoint. In other words we show that the Fong–Tsui conjecture holds for partial isometries, contractive quasi-isometries, or 2-quasi-isometries, and Brownian isometries of positive covariance, or even for a more general class of operators.

Quantization on the Virasoro group

1990

The quantization of the Virasoro group is carried out by means of a previously established group approach to quantization. We explicitly work out the two-cocycles on the Virasoro group as a preliminary step. In our scheme the carrier space for all the Virasoro representations is made out of polarized functions on the group manifold. It is proved that this space does not contain null vector states, even forc≦1, although it is not irreducible. The full reduction is achieved in a striaghtforward way by just taking a well defined invariant subspace ℋ(c, h), the orbit of the enveloping algebra through the vacuum, which is irreducible for any value ofc andh. ℋ(c, h) is a proper subspace of the sp…

Spin-1/2 sub-dynamics nested in the quantum dynamics of two coupled qutrits

2017

In this paper we investigate the quantum dynamics of two spin-1 systems, $\vec{\textbf{S}}_1$ and $\vec{\textbf{S}}_2$, adopting a generalized $(\vec{\textbf{S}}_1+\vec{\textbf{S}}_2)^2$-nonconserving Heisenberg model. We show that, due to its symmetry property, the nine-dimensional dynamics of the two qutrits exactly decouples into the direct sum of two sub-dynamics living in two orthogonal four- and five-dimensional subspaces. Such a reduction is further strengthened by our central result consisting in the fact that in the four-dimensional dynamically invariant subspace, the two qutrits quantum dynamics, with no approximations, is equivalent to that of two non interacting spin 1/2's. The …

Representing compact sets of compact operators and of compact range vector measures

1987

Dynamics of a harmonic oscillator coupled with a Glauber amplifier

2020

A system of a quantum harmonic oscillator bi-linearly coupled with a Glauber amplifier is analysed considering a time-dependent Hamiltonian model. The Hilbert space of this system may be exactly subdivided into invariant finite dimensional subspaces. Resorting to the Jordan-Schwinger map, the dynamical problem within each invariant subspace may be traced back to an effective SU(2) Hamiltonian model expressed in terms of spin variables only. This circumstance allows to analytically solve the dynamical problem and thus to study the exact dynamics of the oscillator-amplifier system under specific time-dependent scenarios. Peculiar physical effects are brought to light by comparing the dynamics…