Executive Summary : | The classification problem for a tuple of commuting isometries is a fundamental problem in operator theory on Hilbert spaces, function theory, and operator algebras. The canonical decomposition of contractions plays a crucial role in various areas of operator algebras and operator theory, such as dilation theory, invariant subspace theory, and operator interpolation problem. The multidimensional Wold-type decomposition of pairs or tuples of commuting isometries (contractions) offers a large class of applications, such as index theory of $C^{*}$-algebras and invariant subspace theory. Slocinski first showed that a pair of doubly commuting isometries has fourfold Wold-type decompositions of the form unitary-unitary, unitary-shift, shift-unitary, and shift-shift. However, previous studies have mostly studied doubly commuting pairs or doubly commuting tuples of contractions on Hilbert spaces. This proposal aims to study the decomposition for various pairs of commuting operators, obtain a complete description of orthogonal spaces for pair and n-tuple of doubly commuting contractions, and provide further evidence of previous results. The concept of numerical radius of an operator has been extensively studied due to their enormous applications in engineering, quantum computing, numerical analysis, and differential equations. The study aims to investigate the numerical radius of multiplicative or sub-multiplicative operators on Hilbert spaces, as well as the bounds for the numerical radius of a pair of doubly commuting isometries, doubly commuting shifts, co-shifts, row isometry, and Crawford number of an operator. |