14.4 Tensor Linear Operator Product Structure

Tensor Linear Operator Product Structure is the organization of the data specific to the operator case of the tensor product of maps, in which both maps of the factor map pair are endomorphisms, producing an induced endomorphism of the tensor product together with the algebraic structure this closure supports.

Specializing the General Structure to Operators

The Endomorphism Factor Map Pair

Where the general structure allows a factor map pair with four possibly distinct spaces, the operator product structure restricts to

so the factor space relation collapses to only two distinct spaces, V and W, each playing the role of both domain and codomain for its own factor.

The Single Ambient Space

Because domain and codomain coincide on each factor, the induced map f tensor g has domain and codomain both equal to V tensor W, giving the operator product structure a single ambient tensor space rather than a domain tensor space and a separate codomain tensor space.

Algebra Generated by the Structure

Closure Under Tensoring

The operator product structure supports an algebra: the set of endomorphisms of V tensor W expressible as f tensor g for some choice of f and g is closed under scalar multiplication of either factor, since

though the set of pure elementary tensor products of operators is not itself closed under addition, since a sum of two such products is generally not expressible as a single tensor product of operators.

Closure Under Composition

The operator product structure is closed under composition, since

again a single tensor product of operators, so this composition remains inside the same family, giving the structure its characteristic closure under the algebra multiplication of the full endomorphism algebra of V tensor W.

Commuting Subalgebra Structure

Two Commuting Embeddings

The operator product structure supports two natural embeddings into the endomorphism algebra of V tensor W: the embedding sending f to f tensor identity, and the embedding sending g to identity tensor g. These two embedded subalgebras commute elementwise,

so the operator product structure decomposes every f tensor g into a product of one operator from each of two mutually commuting subalgebras.

Spectral and Invariant-Subspace Layer

Eigenspace Decomposition

Within the operator product structure, an eigenspace of f tensor g associated to a product eigenvalue lambda times mu decomposes as the sum of tensor products of the lambda-eigenspace of f with the mu-eigenspace of g, taken over every pair of eigenvalues whose product equals the target value, providing an explicit description of the eigenspaces of the induced operator directly from the eigenspaces of the two factor operators.

Invariant Subspace Lattice

The lattice of invariant subspaces of the operator product structure contains, as a distinguished sublattice, every subspace of the form U tensor X where U is invariant under f and X is invariant under g, though the full lattice of invariant subspaces of f tensor g may contain further subspaces not of this product form, particularly when f tensor g possesses repeated eigenvalues arising from distinct pairs of eigenvalues of f and g.