On Semi-weakly n-Hyponormal Weighted Shifts

Semi-weak n-hyponormality is defined and studied using the notion of positive determinant partition. Several examples related to semi-weakly n-hyponormal weighted shifts are discussed. In particular, it is proved that there exists a semi-weakly three-hyponormal weighted shift W (alpha) with alpha (0) = alpha (1) < alpha (2) which is not two-hyponormal, which illustrates the gaps between various weak subnormalities.

Quadratically Hyponormal Weighted Shifts with Recursive Tail

We characterize positive quadratic hyponormality of the weighted shift W-alpha(x) associated to the weight sequence alpha(x) : 1, 1, root x, (root u, root v, root w)(Lambda) with Stampfli recursive tail, and produce an interval in x with non-empty interior in the positive real line for quadratic hyponormality but not positive quadratic hyponormality for such a shift. (C) 2013 Elsevier Inc. All rights reserved.

Backward Extensions of Recursively Generated Weighted Shifts and Quadratic Hyponormality

Given the weight sequence for a subnormal recursively generated weighted shift on Hilbert space, one approach to the study of classes of operators weaker than subnormal has been to form a backward extension of the shift by prefixing weights to the sequence. We characterize positive quadratic hyponormality and revisit quadratic hyponormality of certain such backward extensions of arbitrary length, generalizing earlier results, and also show that a function apparently introduced as a matter of convenience for quadratic hyponormality actually captures considerable information about positive quadratic hyponormality.

Distinctive Convergence in Australian Floral Colours Seen Through the Eyes of Australian Birds

We used a colour-space model of avian vision to assess whether a distinctive bird pollination syndrome exists for floral colour among Australian angiosperms. We also used a novel phylogenetically based method to assess whether such a syndrome represents a significant degree of convergent evolution. About half of the 80 species in our sample that attract nectarivorous birds had floral colours in a small, isolated region of colour space characterized by an emphasis on long-wavelength reflection. The distinctiveness of this 'red arm' region was much greater when colours were modelled for violet-sensitive (VS) avian vision than for the ultraviolet-sensitive visual system. Honeyeaters (Meliphagidae) are the dominant avian nectarivores in Australia and have VS vision. Ancestral state reconstructions suggest that 31 lineages evolved into the red arm region, whereas simulations indicate that an average of five or six lineages and a maximum of 22 are likely to have entered in the absence of selection. Thus, significant evolutionary convergence on a distinctive floral colour syndrome for bird pollination has occurred in Australia, although only a subset of bird-pollinated taxa belongs to this syndrome. The visual system of honeyeaters has been the apparent driver of this convergence.

k-Hyponormality and n-Contractivity for Agler-Type Shifts

We consider k-hyponormality and n-contractivity (k, n = 1, 2, ...) as "weak subnormalities" for a Hilbert space operator. It is known that k-hyponormality implies 2k-contractivity; we produce some classes of weighted shifts including a parameter for which membership in a certain n-contractive class is equivalent to k-hyponormality. We consider as well some extensions of these results to operators arising as restrictions of these shifts, or from linear combinations of the Berger measures associated with the shifts.