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Generalized numerical ranges, dilation, and quantum error correction. (English) Zbl 1435.47011

Botelho, Fernanda (ed.), Recent trends in operator theory and applications. Workshop, The University of Memphis, Memphis, TN, USA, May 3–5, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 737, 25-42 (2019).
This is a survey paper on generalized numerical ranges and their connections to the study of dilation and perturbation of operators. Moreover, applications in quantum error correction and Calkin algebras are introduced.
The authors start by some basic properties of the (classical) numerical range \(W(A)\) of a bounded linear operator \(A\). In particular, they present three algebraic or geometric proofs on the celebrated Toeplitz-Hausdorff theorem which asserts the convexity of \(W(A)\). Though they all reduce the problem to the case of \(2\times 2\) matrices. Besides, interplay between the algebraic properties of operators and the geometric properties of their numerical range is introduced.
A dilation of an operator \(T\) on the Hilbert space \(\mathcal K\) is an operator \(A\) on the Hilbert space \(\mathcal H (\supseteq \mathcal K)\) whose restriction to \(\mathcal K\) with the orthogonal projection onto \(\mathcal H\) is \(T\). The authors discuss the recent development on the studies between dilation of \(T\) to \(I\otimes A\) and the inclusion relation of numerical ranges \(W(T)\subseteq W(A)\). The equivalence holds under certain conditions on \(A\), for example, \(A\) is a \(2\times 2\) matrix or a \(3\times 3\) normal matrix. More general studies on joint dilation and joint numerical range inclusion for \(m\)-tuples of operators \((T_1,\dots,T_m)\) and \((A_1,\dots,A_m)\) are given. Note that the joint numerical range may fail to be convex in general. The inclusion therefore is considered by taking the convex hull. On the other hand, one may apply the adjoint decomposition of operators and focus on adjoint operators \(T_j\) and \(A_j\), \(j=1,\dots,m\). The joint dilation and inclusion are characterized by positivity and complete positivity, respectively, of linear maps between the subspaces spanned by \(\{I,T_1,\dots,T_m\}\) and \(\{I,A_1,\dots,A_m\}\). Therefore, one may study the equivalence in the context of linear maps between two subspaces. Numerical examples are given to illustrate the necessities on the assumptions in the results.
In applications to the existence of quantum error correction codes, joint numerical ranges are generalized to joint higher rank numerical ranges and joint matricial ranges. Both sets capture matrices with special structures which have the joint dilation of an \(m\)-tuple of operators. The authors discuss non-emptiness, convexity and star-shapedness of these sets. In particular, both sets fail to be convex in general, however, they are both non-empty and star-shaped when the dimension of the Hilbert space is sufficiently large. Particularly, when the Hilbert space has infinite dimension, non-emptiness and star-shapedness hold for both sets. In this case, the joint essential matricial range arises when taking the intersection of the joint matricial range of operators varies over a compact perturbation. This concept generalizes the joint essential numerical range, which is studied in the context of Calkin algebra. The authors introduce a convexity result of the joint essential matricial range, which is an extension of the joint essential numerical range. Moreover, connections between the joint essential matricial range and the algebra matricial range are presented.
The authors raise some important and intriguing open problems and explain how these unsolved problems play roles in theoretical and applicable areas. The authors provide a comprehensive survey on how theories and applications have lead to the development of the numerical range in recent years. It is a good reference for researchers and students who would like to explore the topic of numerical range.
For the entire collection see [Zbl 1428.47002].
Reviewer: Tin Yau Tam (Reno)

MSC:

47A12 Numerical range, numerical radius
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A20 Dilations, extensions, compressions of linear operators
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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