Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

Abstract

A. We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed. ITo obtain new bifurcation results, N. Dancer [5] introduced in 1985 a new topological invariant for S 1 -equivariant gradient maps, which provides more information than the usual equivariant one. In 1994 S. Rybicki [14,16] developed the complete degree theory for S 1 -equivariant gradient maps and 3 years later K. Gęba extended this theory to an arbitrary compact Lie group. In 2001 S. Rybicki [15] defined the degree for S 1 -equivariant strongly indefinite functionals in Hilbert space. 10 years later A. Gołębiewska and S. Rybicki [8] generalized this degree to compact Lie groups. The relation between equivariant and equivariant gradient degree theories were studied in [1,2,7].The main goal of this paper is to present a construction and properties of a new degree-type topological invariant Deg ∇ G , which is defined for equivariant gradient perturbations of a equivariant unbounded self-adjoint Hilbert operator with a purely discrete spectrum (in the general case a compact Lie group). As far as we know, the idea of the construction of such an invariant should be attributed to K. Gęba.It is worth pointing out that equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with a purely discrete spectrum appear naturally in a variety of problems in nonlinear analysis, such as the search for periodic solutions of Hamiltonian systems Date: December 24, 2018.

Bifurcations from degenerate orbits of solutions of nonlinear elliptic systems

“…It is worth to point out that the proof relies on the properties of the degree, not its exact definition. Therefore, similar results can be obtained with the use of other degree theories satisfying analogous properties, in particular the property of generalised homotopy invariance (or otopy invariance), see for example [2], [16], [21], [28], [34].…”

Bifurcations from degenerate orbits of solutions of nonlinear elliptic systems

“…It is worth pointing out that the proof relies on the properties of the degree, not its exact definition. Therefore, similar results can be obtained with the use of other degree theories satisfying analogous properties, in particular the property of generalised homotopy invariance (or otopy invariance), see for example [2,16,21,28,34].…”

“…We say that a G-invariant Ω-Morse function ϕ satisfying the assertion of the above lemma is associated with ϕ. Note that from the lemma it follows that ϕ and ϕ are Ω-homotopic, i.e., there exists a G-invariant C 2…”

Generalized Gradient Equivariant Multivalued Maps, Approximation and Degree

“…Compact multivalued perturbations of the identity operator in a Hilbert space will be considered, as well as perturbations of some unbounded self-adjoint operators (comp. [20]). We treat our paper as the preliminary step towards this direction.…”

“…One can think also of the perturbations of unbounded self-adjoint operators in Hilbert spaces; see [20]. We postpone the details to another paper, as well as applications to set-valued variational problems.…”