Normal extensions of subnormal linear relations via quaternionic Cayley transforms
Let \(H\) be a complex Hilbert space and \(U\) a unitary operator. Then \(S=\{ \bigl( (I-U)x,i(I+U)x\bigr):x\in H\}\subseteq H\times H\) is the graph of a linear transformation in \(H\) if and only if \(I-U\) is injective. In this case, the corresponding linear transformation is the inverse Cayley transform of \(U\), which is, in general, an unbounded self-adjoint operator. However, even in the case when \(I-U\) is not injective, the space \(S\) is well defined -- it is a self-adjoint linear relation in \(H\). The concept of Cayley transform has also been studied in the context of quaternions. In the present paper, a similar transform is defined for some linear relations. Namely, to a linear relation \(S\) in \(H^2\), the linear relation \[ V=\{ \bigl( (x_{1}^{\prime},x_{2}^{\prime})+(ix_{1},-ix_2),(x_{1}^{\prime},x_{2}^{\prime})+(-ix_1,ix_2)\bigr):\bigl((x_1,x_{1}^{\prime}),(x_2,x_{2}^{\prime})\bigr)\in S\} \] is associated, which is a quaternionic type Cayley transform. Under some natural conditions, \(V\) is the graph of a partial isometry in \(H^2\). First, some significant properties of unbounded normal operators are recaptured in the framework of normal linear relations and then, elementary properties of quaternionic Cayley transform are discussed. Those linear relations whose quaternionic Cayley transforms are unitary operators are characterized. Since the normal extensions of linear relations which are investigated in this paper are performed via some partial isometries, a part of paper is devoted to the geometric study of mentioned isometries. In the final section, the authors are concerned with the subnormality of some classes of linear relations.
scientific article; zbMATH DE number 6176415