Sigmaizj

Sigmaizj is a hypothetical construct used in operator theory to describe a parameterized family of linear maps on a Hilbert space. The term is formed to reflect a three-part structure: a sigma component and two indices i and j, with an additional complex parameter z that controls a interpolation between subspaces. In this scheme, sigmaizj serves as a bridge between projection-like behavior and more general perturbations, making it useful for didactic illustrations and speculative research.

Definition. Let H be a complex Hilbert space and let {P_i} be a finite family of mutually orthogonal projections with sum equal to the identity. For indices i and j and for z ∈ C, sigma_i,z,j: H → H is defined by sigma_i,z,j(v) = P_i v + z P_j⊥ v, where P_j⊥ = I − P_j. This defines a linear operator that depends holomorphically on z and that preserves the decomposition of H induced by the projections.

Properties. Sigmaizj is linear in its argument and in z. When z = 0, sigma_i,0,j reduces to P_i, the projection onto the i-th subspace. When z is varied, the operator continuously deforms the contribution from the complementary subspace P_j⊥. The adjoint and spectrum of sigma_i,z,j can be analyzed in terms of the projections {P_i} and the parameter z, though explicit formulas depend on the chosen subspace arrangement.

Examples and usage. In a two-projection setting, matrix representations of sigma_i,z,j are readily computed and used to illustrate how small changes in z affect range, null space, and spectral properties. The construction is primarily a teaching tool and a hypothetical model for perturbation phenomena in operator theory.

See also. Projections, operator theory, perturbation theory.