The quaternionic evolution operator
Copyright policy )The quaternionic evolution operator
The authors study quaternionic semigroups and groups generated by quaternionic, in general unbounded, linear operators of the form \(T=T_0+i T_1+j T_2+k T_3\). The components \(T_l\), \(l=0,1,2,3\), do not necessarily commute. The quaternionic version of the Hille-Phillips-Yosida generation theorem is established. This result is based on the fact that the Laplace transform of the quaternionic semigroup \(e^{tT}\) is the \(S\)-resolvent operator \((T^2-2 \text{{Re}[s]}T+| s|^2 J)^{-1} ({\overline s}J-T)\), the quaternionic analogue of the classical resolvent operator. Here, \(J\) denotes the quaternionic identity operator.
Functional calculus for linear operators, Mathematics(all), Quaternionic semigroup, S-spectrum, One-parameter semigroups and linear evolution equations, \(S\)-spectrum, Bounded and unbounded quaternionic generators, S-resolvent operator, Right and left linear quaternionic operators, quaternionic group, \(S\)-resolvent operator, Quaternionic group, quaternionic semigroup, Hille-Phillips-Yosida theorem in the quaternionic setting, Hille–Phillips–Yosida theorem in the quaternionic setting, Spectrum, resolvent, bounded and unbounded quaternionic generators, right and left linear quaternionic operators
Functional calculus for linear operators, Mathematics(all), Quaternionic semigroup, S-spectrum, One-parameter semigroups and linear evolution equations, \(S\)-spectrum, Bounded and unbounded quaternionic generators, S-resolvent operator, Right and left linear quaternionic operators, quaternionic group, \(S\)-resolvent operator, Quaternionic group, quaternionic semigroup, Hille-Phillips-Yosida theorem in the quaternionic setting, Hille–Phillips–Yosida theorem in the quaternionic setting, Spectrum, resolvent, bounded and unbounded quaternionic generators, right and left linear quaternionic operators
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