Complex Unit Disk Research Articles

Distortion coefficients for crypto-unitary operators

For a linear operator T on complex Hilbert space H, we define M ( T) as the least number such that T may be made a contraction by changing to a new norm |·|, derived from an inner product, such that for all h ∈ H ‖ h‖⩽| h|⩽ M( T)‖ h‖.We define K ( T) as the least number such that ‖ p( T)‖⩽ K( T) for all polynomials p mapping the complex unit disk into itself. In this paper it is assumed that T is similar to a unitary and that H is finite-dimensional. We find that the inequalities K ( T) ⩽ M ( T) ⩽ K ( T) 2 always hold, and establish criteria for the equality K ( T) = M ( T). We exhibit an example (with dim H = 12) where K ( T) ≠ M ( T).

Distortion coefficients for cryptocontractions

For complex Hilbert space H of d dimensions and for any number K ⩾ 1, we may define m( K, d) as the least number with the following property: if ‖ p( T)‖ ⩽ K for all polynomials p mapping the complex unit disk into itself, then the operator T may be made a contraction by changing to a new norm |·|, derived from an inner product, such that ‖h‖ ⩽ |h| ⩽m(K,d)‖h‖ (h∈H). It is a long-standing open question whether m( K, d) has a finite bound independent of d. The present paper studies this and related questions and provides, in particular, an explicit estimate for m( K, d)—which, however, grows with d.

A new notion of semicharacters

0. Introduction. A semigroup S is (uniquely) divisible if, for each x E S, and each positive integer n, there exists (a unique) y E S such that yn = x. In the unique case we write y=X1ln. Uniquely divisible commutative semigroups, referred to in the sequel as UDC semigroups, have been characterized in [7]. Compact topological semigroups satisfying this hypothesis have been studied in [6], [11], [12], and [13]. Material of a related nature occurs in [15] and [17]. In [6], the authors showed that if S is a finite-dimensional compact UDC semigroup in which the set of idempotents is totally disconnected, then there exist sufficiently many continuous homomorphisms (semicharacters) of S into the complex unit disk to separate points. This usage of the complex disk as a range space is in line with the classical philosophy, group is to the circle group as Abelian semigroup is to the complex unit disk semigroup. A large amount of work has been done on the investigation of this analogy; in particular, see [9] for a comprehensive survey of the algebraic results in this direction. For the case of Abelian topological semigroups and continuous semicharacters, see [16] and [23]. However, it is clear that the idempotent structure of the complex disk makes it unsuitable as a range space for continuous homomorphisms on general Abelian topological semigroups. In particular, if S is a connected topological semilattice, then any continuous homomorphism of S into the complex disk must be trivial. This deficiency is well known, and reasonable substitutes for the complex disk have been sought for some time. If each maximal group of S is trivial, then one of the most appealing replacements for the disk is some form of thread -a semigroup on a space homeomorphic to the unit interval in which one endpoint acts as an identity and the other as a zero. The complete structure of threads is given in [21], and questions concerning their suitability as range spaces for continuous semicharacters are raised therein. The basic building blocks for threads are U, the interval [0, 1] under real number multiplication; M, the interval [0, 1] under multiplication xy =min {x, y}; and C, the interval [1/2, 1] under multiplication x y = max {1 /2, xy}, where xy represents the ordinary real number product of x and y. The semigroup U has been used successfully as a range space for a certain class of compact UDC semigroups in [6]. The semigroup M is a very logical range space for the category of compact topological semilattices; the problem of whether every