A New Resolvent Equation for the S-Functional Calculus

作者：Alpay Daniel; Colombo Fabrizio; Gantner Jonathan; Sabadini Irene^*

来源：Journal of Geometric Analysis, 2015, 25(3): 1939-1968.

DOI：10.1007/s12220-014-9499-9

• n-tuples of noncommuting operators
• Quaternionic operators
• S-spectrum
• Right S-resolvent operator
• Left S-resolvent operator
• Resolvent equation
• Projectors

The S-functional calculus is a functional calculus for (n + 1)-tuples of not necessarily commuting operators that can be considered a higher-dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S-L(-1) (s, T) and the right one S-R(-1) (s, T), where s = (s(0), s(1),..., s(n)) is an element of Rn+1 and T = (T-0, T-1,..., T-n) is an (n + 1)-tuple of noncommuting operators. The two S-resolvent operators satisfy the S-resolvent equations S-L(-1) (s, T)s - TLS-1 (s, T) = I, and sS(R)(-1) (s, T)-S-R(-1) (s, T) T = I, respectively, where I denotes the identity operator. These equations allow us to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation which is the analog of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.

• 出版日期2015-7