Hwa-Long Gau

Numerical range of s(φ)

We make a detailed study of the numerical ranges W(T) of completely nonunitary contractions T with the property rank (1-T∗T)=1 on a finite-dimensional Hilbert space. We show that such operators are completely characterized by the Poncelet property of their numerical ranges, namely, an n-dimensional contraction T is in the above class if and only if for any point λ on the unit circle there is an (n+l)-gon which is inscribed in the unit circle, circumscribed about W(T) and has λ as a vertex. We also obtain a dual form of this property and the information on the inradii of numerical ranges of arbitrary finite-dimensional operators.

Condition for the numerical range to contain an elliptic disc

For an n-by-n matrix A and an elliptic disc E in the plane, we show that the sum of the number of common supporting lines and the number of common intersection points to E and the numerical range W( A)of A should be at least 2n + 1 in order to guarantee that E be contained in W( A). This generalizes previous results of Anderson and Thompson. As an application, our result is used to verify a special case of the Poncelet property conjecture.

C*-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

. Let V = B(H) or S(H), where B(H) is the algebra of a bounded linear operator acting on the Hilbert space H, and S(H) is the set of self-adjoint operators in B(H). Denote the numerical range of A ∈ B(H) by W(A) = {(Ax,x): x ∈ H, (x,x) = 1}. It is shown that a surjective map O: V → V satisfies W(AB + BA) = W(O(A)O(B) + O(B)O(A)) for all A, B ∈ V if and only if there is a unitary operator U ∈ B(H) such that O has the form X → ±U*XU or X → ±U*X t U, where X t is the transpose of X with respect to a fixed orthonormal basis. In other words, the map O or -O is a C*-isomorphism on B(H) and a Jordan isomorphism on S(H). Moreover, if H has finite dimension, then the surjective assumption on O can be removed.

Line segments and elliptic arcs on the boundary of a numerical range

For an n-by-n complex matrix A, we consider the numbers of line segments and elliptic arcs on the boundary ∂W(A) of its numerical range. We show that (a) if and A has an (n  − 1)-by-(n  − 1) submatrix B with W(B) an elliptic disc, then there can be at most 2n − 2 line segments on ∂W(A), and (b) if , then ∂W(A) contains at most (n  − 2) arcs of any ellipse. Moreover, both upper bounds are sharp. For nilpotent matrices, we also obtain analogous results with sharper bounds. §Dedicated to Ky Fan on his ninety-third birthday. ¶Part of the results here was presented by the second author in the 8th WONRA at Bremen, Germany and the 13th ILAS Conference at Amsterdam, the Netherlands in July, 2006.

Dilation to inflations of s(φ)

Let Tbe a completely nonunitary contraction with rank (l− T ∗ T)=l on an n-dimensional Hilbert space. We prove that (1) if n= 2 and Sis an operator which has norm 1, attains its norm and satisfies W( S) ⊆ W(T), then Shas Tas a direct summand, and (2) if n≥ 3 and Sis an operator such that Sk dilates to simultaneously for k=l,2,…,n−1 and then Shas Tas a direct summand. (Here W(·) denotes the numerical range). These results generalize the corresponding ones for T the n× nnilpotent Jordan block.

Higher Rank Numerical Ranges of Normal Matrices

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix

Weighted shift matrices: Unitary equivalence, reducibility and numerical ranges

A\in M_n

Finite Blaschke products of contractions

has eigenvalues

Numerical Range of a Normal Compression

a_1,\dots,a_n

Product of operators and numerical range

, then its higher rank numerical range