Narrow Results

Let T be a bounded linear operator on an infinite dimensional, separable Banach space X. We consider a class of supercyclic operators, whose point spectrum of their adjoints are nonempty, and prove that under certain conditions, the orbit of every supercyclic vector for such an operator is unbounded. This result has some nice applications: (1) We obtain some conditions equivalent to the supercyclicity of extreme points of the closed unit ball of on a separable infinite dimensional Hilbert space; this helps us to characterize all supercyclic operators in this class. (2) The adjoint of composition operators on certain weighted Hardy spaces are never supercyclic. Next, we turn our attention to the commutant and show that if T is an operator in the mentioned class, then every operator in the commutant of T is not hypercyclic.

In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of 𝔻 such that φ(ζ) and ψ(ζ) belong to ∂𝔻 for some ζ ∈ ∂𝔻 and u, v ∈ H^∞ are continuous on ∂𝔻 with u(ζ)v(ζ) ≠ 0, then is compact on H² if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H². We verify that if φ is an analytic self-map of 𝔻 with Denjoy–Wolff point b ∈ 𝔻 and u ∈ H^∞\{0}, then every weighted composition operator in the commutant {W_{u, φ}}′ has {f ∈ H² : f(b) = 0} as its nontrivial invariant subspace.

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that H_n(q), U_q(sl₂) act as a dual pair on the tensored vector representation for all q.

In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial.

We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on ${ℬ}(h)$, not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible $(H,\beta )$-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.

We obtain all Dirichlet spaces ℱ_q, q ∈ ℝ, of holomorphic functions on the unit ball of ℂ^N as weighted symmetric Fock spaces over ℂ^N. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱ_q. We pick out the analytic Hilbert modules from among the ℱ_q. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱ_q. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱ_q fit in exact sequences that are in the same Ext class. We identify the groups K₀ and K₁ of the Toeplitz algebras on the ℱ_q arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱ_b that depend on a weight sequence b.

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra8 (1968) 415–417] proved that a nilpotent Moufang loop is the direct product of $p$-loops for some primes $p$, consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence.