On the Structure of Unbounded Linear Operators
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RESEARCH PAPER
On the Structure of Unbounded Linear Operators Shahrzad Azadi1 • Mehdi Radjabalipour2,3 Received: 18 June 2020 / Accepted: 9 August 2020 Ó Shiraz University 2020
Abstract Given a densely defined closed operator T : DðTÞ H ! K, von Neumann defined W :¼ ðI þ T TÞ1 and showed that 0 W I, KðWÞ ¼ f0g and T T ¼ ðI WÞW 1 ¼ W 1 ðI WÞ with DðT TÞ ¼ RðWÞ. (Here, DðÞ, RðÞ, KðÞ and, later, GðTÞ stand for the domain, the range, the kernel, and the graph of a linear transformation, respectively.) The functional calculus is not applicable, in general, to guess a formula like ðI WÞ1=2 W 1=2 for jTjð:¼ ðT TÞ1=2 Þ and to achieve a polar decomposition T ¼ VjTj for T. Also, the operators T and T do not exist as single-valued operators to be able to define W and extend our conjectures to arbitrary unbounded linear operators. The task of the present paper is to define the von Neumann operator W directly from GðTÞ and prove all the desired extensions. Keywords Unbounded linear operator Closure of an operator von Neumann generator Adjoint of an operator Absolute value Mathematics Subject Classification 47A06 47B15 47B25
1 Introduction The notion of unbounded operators was first introduced by von Neumann (1930) to base quantum mechanics on a solid mathemathical foundation. The present paper studies the structure and certain properties of unbounded (singlevalued) linear operators T between Hilbert spaces. The work extends the representation T T ¼ ðI WÞW 1 with DðT TÞ ¼ RðWÞ known for densely defined closed operators, where W ¼ ðI þ T TÞ1 I is an injective positive operator. We will show that jTj :¼ ðT TÞ1=2 is a well-defined unbounded positive operator represented as jTj ¼ ðI WÞ1=2 W 1=2 with DðjTjÞ ¼ RðW 1=2 Þ. Also, T is & Mehdi Radjabalipour [email protected]; [email protected] Shahrzad Azadi [email protected] 1
Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran
2
The Iranian Academy of Sciences, Haghany Highway, National Library Exit, Tehran, Iran
3
Department of Mathematics, Sh. B. University of Kerman, Kerman, Iran
shown to have a polar decomposition T ¼ VjTj with DðTÞ ¼ DðjTjÞ, where V 2 BðH; KÞ is an isometry. The importance of such a result follows from the fact that not every product of the form f ðWÞgðWÞ1 is selfadjoint to which the extended functional calculus be applied; for instance, if f ðtÞ ¼ gðtÞ t and if W is the multiplication by f on L2 ð½0; 1Þ, then f ðWÞgðWÞ1 ¼ IjRðWÞ 6¼ I ¼ gðWÞ1 f ðWÞ ¼ ½f ðWÞgðWÞ1 . The results are further extended to general unbounded (single-valued) linear operators whose closures or adjoints as defined below may not be necessarily single-valued. Recall that a multilinear operator T between Hilbert spaces H and K is any linear subspace of H K; the subspace maybe denoted by the letter T itself; when T is viewed as an action, the notation GðTÞ, called the graph of T, is preferred. The domain DðTÞ, the range RðTÞ, and the
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