The notion of an essentially slant Toeplitz operator on the space L is introduced and some of the properties of the set ESTO(L), the set of all essentially slant Toeplitz operators on L, are investigated. In particular the conditions under which the product of two operators in ESTO(L) is in ESTO(L) are discussed. The notion is generalized to kth-order essentially slant Toeplitz operators. The n...

In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipsc...

Consider a Hilbert space obtained as the completion of the polynomials C[z ] in m-variables for which the monomials are orthogonal. If the commuting weighted shifts defined by the coordinate functions are essentially normal, then the same is true for their restrictions to invariant subspaces spanned by monomials. This generalizes the result of Arveson [4] in which the Hilbert space is the m-shi...

In this paper I offer a defence of a Russellian analysis of the referential uses of incomplete (mis) descriptions, in a contextual setting. With regard to descriptions (introduction), I will support the former against the latter. In 1. I explain what I mean by “essentially” incomplete descriptions: incomplete descriptions are context dependent descriptions. In 2. I examine one of the best versi...

Although sensory perception and neurobiology are traditionally investigated one modality at a time, real world behaviour and perception are driven by the integration of information from multiple sensory sources. Mounting evidence suggests that the neural underpinnings of multisensory integration extend into early sensory processing. This article examines the notion that neocortical operations a...

In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]dimensional subspace which has further proportionally dimensional Euclidean subspaces of any proportion. We consider a space X1 = (Rn, ‖ · ‖1) with the property that if a space X2 = (Rn, ‖ · ‖2) is “no...

For f in L∞, the space of essentially bounded Lebesgue measurable functions on the unit circle, ∂D, the Toeplitz operator with symbol f is the operator Tf on the Hardy space H2 of the unit circle defined by Tfh = P (fh). Here P denotes the orthogonal projection in L2 with range H2. There are many fascinating problems about Toeplitz operators ([3], [6], [7] and [20]). In this paper we shall conc...

The Lake: Alice, Bertha, and Claudia live around Forest Lake, at some distance from each other. The ecosystem is sensitive, in particular to the solvent they have been using when painting their boats, and the lake is faring badly. Until recently, they had no way of knowing that the solvent would have this effect, but they just learned that it is in fact killing the fish in the lake, and that th...