@MISC{Szafraniec906normals,subnormals, author = {Franciszek Hugon Szafraniec}, title = {Normals, subnormals and an open question}, year = {906} }

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Abstract

Abstract. An acute look at basic facts concerning unbounded subnormal operators is taken here. These operators have the richest structure and are the most exciting among the whole family of beneficiaries of the normal ones. Therefore, the latter must necessarily be taken into account as the reference point for any exposition of subnormality. So as to make the presentation more appealing a kind of comparative survey of the bounded and unbounded case has been set forth. This piece of writing serves rather as a practical guide to this largely impenetrable territory than an exhausting report. We begin with bounded operators pointing out those well known properties of normal and subnormal operators, which in unbounded case become much more complex. Then we are going to show how the situation looks like for their unbounded counterparts. The distinguished example of the creation operator coming from the quantum harmonic oscillator crowns the theory. Finally we discuss an open question, one of those which seem to be pretty much intriguing and hopefully inspiring. By an unbounded operator we mean a not necessarily bounded one, nevertheless it is always considered to be densely defined, always in a complex Hilbert space. If we want to emphasis an operator to be everywhere defined we say it is on, otherwise we say it is in. Unconventionally though suggestively, B(H) denotes all the bounded operators on H. If A is an operator, then D(A), N(A) and R(A) stands for its domain, kernel(null space) and range respectively; if A is closable, its closure is denoted by A.