Do you want to publish a course? Click here

Lifting Commuting 3-Isometric Tuples

49   0   0.0 ( 0 )
 Added by Benjamin Russo
 Publication date 2015
  fields
and research's language is English




Ask ChatGPT about the research

An operator $T$ is called a 3-isometry if there exists operators $B_1(T^*,T)$ and $B_2(T^*,T)$ such that [Q(n)=T^{*n}T^n=1+nB_1(T^*,T)+n^2 B_2(T^*,T)] for all natural numbers $n$. An operator $J$ is a Jordan operator of order $2$ if $J=U+N$ where $U$ is unitary, $N$ is nilpotent order $2$, and $U$ and $N$ commute. An easy computation shows that $J$ is a $3$-isometry and that the restriction of $J$ to an invariant subspace is also a $3$-isometry. Those $3$-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely positive maps, in terms of a positivity condition on the operator pencil $Q(s).$ In this article, we establish the analogous result in the multi-variable setting and show, by modifying an example of Choi, that an additional hypothesis is necessary. Lastly we discuss the joint spectrum of sub-Jordan tuples and derive results for 3-symmetric operators as a corollary.



rate research

Read More

The characteristic function has been an important tool for studying completely non unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space $clh$. We show that the characteristic function, which is now an operator valued analytic function on the open Euclidean unit ball in $mathbb{C}^n$, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.
116 - Nico Spronk 2015
We determine when contractive idempotents in the measure algebra of a locally compact group commute. We consider a dynamical version of the same result. We also look at some properties of groups of measures whose identity is a contactive idempotent.
For a commuting $d$- tuple of operators $boldsymbol T$ defined on a complex separable Hilbert space $mathcal H$, let $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ be the $dtimes d$ block operator $big (!!big (big [ T_j^* , T_ibig ]big )!!big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, ldots, T_d,T_d^*)$ introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting $d$ - tuple of $d$ - normal operators, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ must be $0$. We show that if the $d$- tuple $boldsymbol T$ is cyclic, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ is non-negative and the compression of a fixed set of words in $T_j^* $ and $T_i$ -- to a nested sequence of finite dimensional subspaces increasing to $mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $big [!! big [ boldsymbol T^* , boldsymbol Tbig ] !!big ]$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting $d$ - tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
81 - David Herrera 2020
Lins theorem states that for all $epsilon > 0$, there is a $delta > 0$ such that for all $n geq 1$ if self-adjoint contractions $A,B in M_n(mathbb{C})$ satisfy $|[A,B]|leq delta$ then there are self-adjoint contractions $A,B in M_n(mathbb{C})$ with $[A,B]=0$ and $|A-A|,|B-B|<epsilon$. We present full details of the approach in arXiv:0808.2474, which seemingly is the closest result to a general constructive proof of Lins theorem. Constructive results for some special cases are presented along with applications to the problem of almost commuting matrices where $B$ is assumed to be normal and also to macroscopic observables.
88 - M. Dawson , G. Olafsson , 2018
On the unit ball B^n we consider the weighted Bergman spaces H_lambda and their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of widetilde{SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of $widetilde{SU(n,1)}$, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of widetilde{SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly develop the restriction principle for each (conjugacy class of) maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform. In particular, we obtain a multiplicity one result for the restriction of the holomorphic discrete series to all maximal abelian subgroups. We also observe that the Segal-Bargman transform is (up to a unitary transformation) a convolution operator against a function that we write down explicitly for each case. This can be used to obtain the explicit simultaneous diagonalization of Toeplitz operators whose symbols are invariant by one of these maximal abelian subgroups.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا