Continuous Slice Functional Calculus in Quaternionic Hilbert Spaces

Abstract

Abstract. The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C * -algebras and to define, on each of these C * -algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

Weyl-von Neumann-Berg theorem for quaternionic operators

“…Proof. First, observe that, if N + ∈ B(H Jm + ) is the unique operator such that N = N + , then the spectrum σ(N + ) is σ(N + ) = Ω K ∩ C + m = K (see [1,Corollary 5.13] for details). Hence by [3,Theorem 3], N + = D + + K + , where D + is a diagonal operator, K + is a Hilbert-Schmidt operator on H Jm + .…”

“…In particular, if T ∈ B(H) is normal, there exists an anti self-adjoint, unitary J ∈ B(H) such that T J = JT (see [1,Theorem 5.9] for details). Hence in this case all the statements in Theorem 2.3 holds true.…”

“…This can be done if there exists an anti self-adjoint unitary operator which commutes with the given operator. There always exists such an operator in case if the operator is normal (see [1,Theorem 5.9, Proposition 3.11] for details for the case of bounded operators). The unbounded case is discussed in [5] in a general setting.…”

S-spectrum and associated continuous frames on quaternionic Hilbert spaces

“…For quaternions and quaternionic Hilbert spaces we refer the reader to [1]. We shall use the recently introduced S-spectral calculus of quaternionic operators for which we refer to [12,6,5,4]. …”

“…Unless q is real the scalar multiple qA of a right linear operator is not necessarily right linear and several other usual properties of a scalar multiple of an operator may not hold [12,17,16]. The adjoint A † of A is defined as…”