Spectral radius and infinity norm of matrices

Let Mn(R) be the linear space of all n×n matrices over the real field R. For any A∈Mn(R), let ρ(A) and ‖A_‖∞ denote the spectral radius and the infinity norm of A, respectively. By introducing a class of transformations φa on Mn(R), we show that, for any A∈Mn(R), ρ(A)<‖A_‖∞ if . If A∈Mn(R) is nonnegative, we prove that ρ(A)<‖A_‖∞ if and only if , and ρ(A)=‖A_‖∞ if and only if the transformation φ‖A_‖∞ preserves the spectral radius and the infinity norm of A. As an application, we investigate a class of linear discrete dynamic systems in the form of X(k+1)=AX(k). The asymptotical stability of the zero solution of the system is established by a simple algebraic method.     

A pattern sequence approach to Stern’s sequence

Suppose that w∈1{0,1}^∗ and let a_w(n) be the number of occurrences of the word w in the binary expansion of n. Let {s(n)}_n?0 denote the Stern sequence, defined by s(0)=0, s(1)=1, and for n?1, In this note, we show that where denotes the complement of w (obtained by sending 0?1 and 1?0) and [w]₂ denotes the integer specified by the word w∈{0,1}^∗ interpreted in base 2.     

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Numerical ranges of weighted shifts

Let A be a unilateral (resp., bilateral) weighted shift with weights wn, n?0 (resp., −∞<n<∞). Eckstein and Rácz showed before that A has its numerical range W(A) contained in the closed unit disc if and only if there is a sequence (resp., ) in [−1,1] such that ²|wn|=(1−an)(1+an+1) for all n. In terms of such an?s, we obtain a necessary and sufficient condition for W(A) to be open. If the wn?s are periodic, we show that the an?s can also be chosen to be periodic. As a result, we give an alternative proof for the openness of W(A) for an A with periodic weights, which was first proven by Stout. More generally, a conjecture of his on the openness of W(A) for A with split periodic weights is also confirmed.     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).     

On hyper-singular integral operators with variable kernels

Let n?2, Sn−1 be the unit sphere in Rn. For 0?α<1, m∈N₀, 1<p?2, and Ω∈L^∞(Rn)×Hr(Sn−1) with (where Hr is the Hardy space if r?1 and Hr=Lr if 1<r<∞), we study the singular integral operator, for r?1, defined by

     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).     

Orlicz spaces without extreme points

Orlicz function and sequence spaces unit balls of which have no extreme points are completely characterized for both (the Orlicz and the Luxemburg) norms. Their subspaces of order continuous elements, with the norms induced from the whole Orlicz spaces without extreme points in their unit balls are also characterized. The well-known spaces L₁ and c₀ with unit balls without extreme points are covered by our results. Moreover, a new example of a Banach space without extreme points in its unit ball is given (see Example 1). This is the subspace a(L₁+L_∞) of order continuous elements of the space L₁+L_∞ equipped with the norm whenever 0<a<∞ and μ(T)>1/a.