Dilation theory is a paradigm for studying operators by way of exhibiting an operator as compression another which in some sense well behaved. For example, every contraction can be dilated to (i.e., of) unitary operator, and on this simple fact penetrating non-normal has been developed. In the first part survey, I will leisurely review key classical results dilation single or several commuting ...

Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing to obtain joint dilations of the families of operators satisfying certain commutation relations encoded by the graph structure. Several applications to the analysis of the structure of concrete graph operator algebras are presented. The starting point of classical dilation theory was the construction of...

Mathematical Morphology (MM) is a tool that can be applied to many digital image processing tasks include the reduction of impulsive or salt and pepper noise in grayscale images. The morphological filters used for this task are resulting from two basic operators: erosion dilation. However, when level contamination higher, these tend distort image. In work we propose pair operators with properti...

The fractional Laplacian (−�)γ/2 commutes with the primary coordination transformations in the Euclidean space Rd: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential Iγ which is dilationinvariant and translation-invariant. In this work, we investigate the functional properties (contin...

The area of knowledge merging is concerned with merging conflicting information while preserving as much as possible. Most proposals in the literature work with knowledge bases expressed in propositional logic. We propose a new framework for merging knowledge bases expressed in (subsets of) first-order logic. Dilation operators (a concept originally introduced by Bloch and Lang) are employed an...

We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson [6] and Gabardo and Han [5]. We also extend the dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun [4], by ide...