Partial Isometries on a Hilbert Space

An isometry on a normed linear space, our context is a Hilbert space, is a norm preserving linear map, or operator. A partial isometry is an isometry defined on a subspace which maps the orthogonal complement to zero. Over the past century these operators have played a fundamental role in the study of operators, and of algebras of operators, through for example the polar decomposition of an operator, the classification of projections and von Neumann algebras, and the K-theory of operator algebras. The algebras generated by a unitary operator, or an isometry, are well understood. In joint work with Z. Niu we establish a close connection between partial isometries and contractions to examine the algebra generated by a partial isometry.