Spectral multiplicity and odd \(K\)-theory (Q547845)

In this paper, the authors begin a study on families of unbounded self-adjoint Fredholm operators on a separable Hilbert space with compact resolvent, which can be converted to families of bounded self-adjoint Fredholm operators by functional calculus. The contents are as follows: 2. The multiplicity of eigenvalues; 3. Spectral exhaustions and spectral flow; 4. Families with spectrum of constant multiplicity; 5. Families with bounded multiplicity; 6. Multiplicity less than or equal to two; 7. Concluding remarks. The main results are: Theorem 3.6. A spectral exhaustion exists for the family of regular, unbounded self-adjoint Fredholm operators over a compact Hausdorff space \(X\) if and only if the spectral flow of the family, that is, the cohomology class of an associated cocycle on \(X\), is zero. Corollary 4.4. For a family with spectrum of constant multiplicity, if the spectral flow of the family is zero, then the class of the family is trivial in the topological odd K-theory group \(K^1(X)\), which is viewed as the homotopy set from \(X\) to the space of bounded self-adjoint Fredholm operators with essential spectrum both positive and negative. Theorem 6.4. For a family of regular, unbounded self-adjoint Fredholm operators on the odd-dimensional sphere with dimension greater than or equal to five, such as in the case where a certain three-dimensional cohomology class vanishes, if the spectral flow of the family is zero and the multiplicity of eigenvalues is less than or equal to two, then the class of the family is rationally trivial in K-theory. The last theorem tells us that the multiplicity of eigenvalues has an effect on the classification of families of unbouded self-adjoint Fredholm operators.