Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1 = (c+ c )/2 and b2 = (c− c)/2i. Also write B for the spectral scale of {b1, b2} relative to τ . In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information ...

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius √ 2. Since the spectrum of non-hermitian random...

Stampfli has shown that for a given T £ B(H) there exists a K £ C(H) so that o(T + K) = ow(T). An analogous result holds for the essential numerical range We(T). A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator T £ B(H) if o(T + K) = o„(T) and W(T + K)= We(T). Theorem. For each block diagonal operator T, there exists a compact operator K wh...

Motivated by applications in quantum information and quantum control, a new type of C-numerical range, the relative C-numerical range denoted W K (C, A), is introduced. It arises upon replacing the unitary group U (N) in the definition of the classical C-numerical range by any of its compact and connected subgroups K ⊂ U (N). The geometric properties of the relative C-numerical range are analys...