TY - JOUR

T1 - Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

AU - Shamovich, Eli

AU - Vinnikov, Victor

N1 - Funding Information:
The research of E.S. was partially carried out during the visits to the Department of Mathematics and Statistics of the University of Konstanz, supported by the EDEN Erasmus Mundus program (30.12.2013–30.6.2014) and to the MFO, supported by the Leibnitz graduate student program (6.4.2014–12.4.2014). The research of E.S. was also supported by the Negev fellowship of the Kreitman school of the Ben Gurion University of the Negev.
Publisher Copyright:
© 2016

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We study non-skewselfadjoint representations of a finite dimensional real Lie algebra g. To this end we embed a non-skewselfadjoint representation of g into a more complicated structure, that we call a g-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group G. We develop the frequency domain theory of the system in terms of representations of G, and introduce the joint characteristic function of a g-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the ax+b group, the group of affine transformations of the line.

AB - We study non-skewselfadjoint representations of a finite dimensional real Lie algebra g. To this end we embed a non-skewselfadjoint representation of g into a more complicated structure, that we call a g-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group G. We develop the frequency domain theory of the system in terms of representations of G, and introduce the joint characteristic function of a g-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the ax+b group, the group of affine transformations of the line.

KW - Non-selfadjoint operators and characteristic functions

KW - Overdetermined multidimensional systems (operator vessels)

KW - Representations of Lie algebras and Lie groups

KW - Taylor joint spectrum

UR - http://www.scopus.com/inward/record.url?scp=85055967595&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2016.08.001

DO - 10.1016/j.jfa.2016.08.001

M3 - Article

AN - SCOPUS:85055967595

VL - 276

SP - 1

EP - 44

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -