TY - JOUR

T1 - A New Resolvent Equation for the S-Functional Calculus

AU - Alpay, Daniel

AU - Colombo, Fabrizio

AU - Gantner, Jonathan

AU - Sabadini, Irene

N1 - Publisher Copyright:
© 2014, Mathematica Josephina, Inc.

PY - 2015/7/20

Y1 - 2015/7/20

N2 - The $$S$$S-functional calculus is a functional calculus for (Formula presented.)-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 (Formula presented.) and the right one (Formula presented.), where (Formula presented.) and (Formula presented.) is an (Formula presented.)-tuple of noncommuting operators. The two S-resolvent operators satisfy the S-resolvent equations (Formula presented.), and (Formula presented.), respectively, where (Formula presented.) 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.

AB - The $$S$$S-functional calculus is a functional calculus for (Formula presented.)-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 (Formula presented.) and the right one (Formula presented.), where (Formula presented.) and (Formula presented.) is an (Formula presented.)-tuple of noncommuting operators. The two S-resolvent operators satisfy the S-resolvent equations (Formula presented.), and (Formula presented.), respectively, where (Formula presented.) 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.

KW - Left s-resolvent operator

KW - Projectors

KW - Quaternionic operators

KW - Resolvent equation

KW - Right s-resolvent operator

KW - n-tuples of noncommuting operators

KW - s-spectrum

UR - http://www.scopus.com/inward/record.url?scp=84934985449&partnerID=8YFLogxK

U2 - 10.1007/s12220-014-9499-9

DO - 10.1007/s12220-014-9499-9

M3 - Article

AN - SCOPUS:84934985449

VL - 25

SP - 1939

EP - 1968

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 3

ER -