Pei Yuan Wu

THE OPERATOR FACTORIZATION PROBLEMS

We survey various results concerning operator factorization problems. More precisely, we consider the following setting. Let H be a complex Hilbert space, and let .G?( H) be the algebra of all bounded linear operators on H. For a given subset % of 3?(H), we are interested in the characterization of operators in Q(H) which are expressible as a product of finitely many operators in V and, for each such operator, the minimal number of factors in a factorization. The classes of operators we consider include normal operators, involutions, partial isometries together with their various subclasses, and other miscellaneous classes of operators. Most of the known results are for operators on finite-dimensional spaces or finite matrices. The paper concludes with some applications, due to Hochwald, concerning the uniqueness of the adjoint operation on operators.

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.

Additive combinations of special operators

This is a survey paper on additive combinations of certain special-type operators on a Hilbert space. We consider (finite) linear combinations, sums, convex combinations and/or averages of operators from the classes of diagonal operators, unitary operators, isometries, projections, symmetries, idempotents, square-zero operators, nilpotent operators, quasinilpotent operators, involutions, commutators, self-commutators, norm-attaining operators, numerical-radius-attaining operators, irreducible operators and cyclic operators. In each case, we are mainly concerned with the characterization of such combinations and the minimal number of the special operators required in them. We will omit the proofs of most of the results here but give some indication or brief sketch of the ideas behind and point out the remaining open problems.

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.

Sums of idempotent matrices

Abstract We show that any complex square matrix T is a sum of finitely many idempotent matrices if and only if trT is an integer and trT ⩾ rank T. Moreover, in this case the idempotents may be chosen such that each has rank one and has range contained in that of T. We also consider the problem of the minimum number of idempotents needed to sum to T and obtain some partial results.

The numerical range of a nonnegative matrix

Abstract We offer an almost self-contained development of Perron–Frobenius type results for the numerical range of an (irreducible) nonnegative matrix, rederiving and completing the previous work of Issos, Nylen and Tam, and Tam and Yang on this topic. We solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin. Some related results are obtained and some open problems are also posed.

Numerical range of Aluthge transform of operator

For any operator A on a Hilbert space, let A denote its Aluthge transform. In this paper, we prove that the closure of the numerical range of A is always contained in that of A. This supplements the recently proved case for dimkerA⩽dimkerA* by Yamazaki, and partially confirms a conjecture of Jung, Ko and Pearcy.

Products of positive semidefinite matrices

Abstract We characterize the complex square matrices which are expressible as the product of finitely many positive semidefinite matrices; a matrix T can be expressed as such if and only if det T⩾0; moreover, the number of factors can always be limited to five. We also determine those matrices which can be expressed as the product of two or four positive semidefinite matrices. These results are analogous to the ones obtained before by C.S. Ballantine for products of positive definite matrices.

Products of Nilpotent Matrices

We show that any complex singular square matrix T is a product of two nilpotent matrices A and B with rank A = rank B = rank T except when T is a 2 X 2 nilpotent matrix of rank one. An n X n complex matrix T is nilpotent if T” = 0. It is easily seen that a product of finitely many nilpotent matrices must be singular. The purpose of this note is to prove the converse.

CONVEX COMBINATIONS OF PROJECTIONS

Abstract On an n -dimensional inner-product space, every operator T that satisfies O ⩽ T ⩽ I is a convex combination of as few as [log 2 n ] + 2 projections, and this number is sharp. If O ⩽ T ⩽ I and trace T is a rational number, then T is an average of projections. Further results are also obtained for the cases when the projections are required to have the same rank and⧸or to be commuting. In each case, the optimal number of projections is determined.