Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComplex Free Spectrahedra, Absolute Extreme Points, and Dilations
62
0
0.0
(
0
)
Authors
Benjamin Passer
Ask ChatGPT about the research
No Arabic abstract
Evert and Helton proved that real free spectrahedra are the matrix convex hulls of their absolute extreme points. However, this result does not extend to complex free spectrahedra, and we examine multiple ways in which the analogous result can fail. We also develop some local techniques to determine when matrix convex sets are not (duals of) free spectrahedra, as part of a continued study of minimal and maximal matrix convex sets and operator systems. These results apply to both the real and complex cases.
rate research
Read More
We continue the study dilation of linear maps on vector spaces introduced by Bhat, De, and Rakshit. This notion is a variant of vector space dilation introduced by Han, Larson, Liu, and Liu. We derive vector spa
The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in {em The functional and harmonic analysis of wavelets and frames} (San Antonio, TX, 1999), {em Contemp. Math}. {bf 247} (1999), 149-182 as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {em theor`eme principal} of B. Sz-Nagy go much beyond it.
We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $|Id + T^2|=1 + |T^2|$ holds for every bounded linear operator $T:Xlongrightarrow X$. This answers in the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for operators, emph{Indiana U. Math. J.} textbf{56} (2007), 2385--2411]. More concretely, we show that this is the case of some $C(K)$ spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, emph{Math. Ann.} textbf{330} (2004), 151--183] and [Plebanek, A construction of a Banach space $C(K)$ with few operators, emph{Topology Appl.} textbf{143} (2004), 217--239]. We also construct compact spaces $K_1$ and $K_2$ such that $C(K_1)$ and $C(K_2)$ are extremely non-complex, $C(K_1)$ contains a complemented copy of $C(2^omega)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $ell_infty$.
We describe here the higher rank numerical range, as defined by Choi, Kribs and Zyczkowski, of a normal operator on an infinite dimensional Hilbert space in terms of its spectral measure. This generalizes a result of Avendano for self-adjoint operators. An analogous description of the numerical range of a normal operator by Durszt is derived for the higher rank numerical range as an immediate consequence. It has several interesting applications. We show using Durszts example that there exists a normal contraction $T$ for which the intersection of the higher rank numerical ranges of all unitary dilations of $T$ contains the higher rank numerical range of $T$ as a proper subset. Finally, we strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it.
Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
