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Register a new userHigher Rank Numerical Ranges of Normal Operators and unitary dilations
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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.
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The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.
We give a brief introduction to private quantum codes, a basic notion in quantum cryptography and key distribution. Private code states are characterized by indistinguishability of their output states under the action of a quantum channel, and we show that higher rank numerical ranges can be used to describe them. We also show how this description arises naturally via conjugate channels and the bridge between quantum error correction and cryptography.
This note is to indicate the new sphere of applicability of the method developed by Mlak as well as by the author. Restoring those ideas is summoned by current developments concerning $K$-spectral sets on numerical ranges.
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.
Let $A$ be a normal operator in a Hilbert space $mathcal{H}$, and let $mathcal{G} subset mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $mathcal{G}$ , and $L$ that makes the system of iterations ${A^ng: gin mathcal{G},;0leq n< L(g)}$ complete, Bessel, a basis, or a frame for $mathcal{H}$. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.
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