14.2.1 Tensor Linear Operator Product Area

Explore how tensor linear operators interact in product spaces, defining key structures in multilinear algebra and tensor calculus.

Tensor Linear Operator Product Area is the branch of study concerned with tensor products of linear maps in the special case where the maps involved are operators, meaning linear maps from a vector space to itself, and with the algebraic and spectral structure that this specialization produces.

Distinguishing the Operator Case

Endomorphisms as the Common Setting

Where the general tensor product of maps allows f and g to act between unrelated pairs of spaces, the operator product area restricts attention to

f : V \to V

g : W \to W

so that the induced map

f \otimes g : V \otimes W \to V \otimes W

is again an endomorphism of a single space, namely the tensor product V tensor W, rather than a map between two different spaces. This closure under tensoring is what allows an algebra of operators, rather than merely a collection of maps, to be built from the construction.

Algebra Structure Induced on the Tensor Product

Because both the domain and codomain of f tensor g coincide, the set of operators on V tensor W obtained this way inherits a multiplication from operator composition, and the operator area studies how the subalgebra generated by all such tensor products of operators sits inside the full algebra of endomorphisms of V tensor W.

Composition Within the Operator Area

The Product Rule for Operators

For operators, the general composition identity for tensor products of maps specializes to

(f_{1} \otimes g_{1}) \circ (f_{2} \otimes g_{2}) = (f_{1} \circ f_{2}) \otimes (g_{1} \circ g_{2}),

now composing two operators on V tensor W to produce a third operator on V tensor W, so that the operator area is closed under both tensoring and composing without leaving the algebra of endomorphisms of the tensor product.

Commutativity of the Two Factor Actions

A central feature of the operator area is that the action coming from the first factor and the action coming from the second factor commute with each other,

(f \otimes {id}_{W}) \circ ({id}_{V} \otimes g) = f \otimes g = ({id}_{V} \otimes g) \circ (f \otimes {id}_{W}) .

This commuting-actions phenomenon is what gives the operator area its characteristic structure, since it means the two families of extended operators, one for each factor, form commuting subalgebras inside the full endomorphism algebra of V tensor W.

Spectral Theory Within the Operator Area

Product Eigenvalue Structure

The spectral theory belonging to this area rests on the fact that eigenvalues of f tensor g are exactly the pairwise products of an eigenvalue of f and an eigenvalue of g, with eigenvector u tensor w whenever u is an eigenvector of f and w is an eigenvector of g. This gives the full spectrum of the tensor product operator as a product set of the two individual spectra, counted with multiplicity equal to the product of the individual multiplicities.

Diagonalizability

If f and g are each diagonalizable, then f tensor g is diagonalizable, since a basis of eigenvectors for V tensor W can be built from elementary tensors of eigenvectors of f and eigenvectors of g, and this basis diagonalizes f tensor g simultaneously by the product eigenvalue rule above.

Invariant Subspace Structure

Products of Invariant Subspaces

If a subspace U of V is invariant under f and a subspace X of W is invariant under g, then the subspace U tensor X of V tensor W is invariant under f tensor g, since the defining rule sends elementary tensors of U tensor X to elementary tensors again lying in U tensor X. This gives a systematic way of building invariant subspaces of the tensor product operator directly from invariant subspaces of the two factor operators, without needing to search for invariant subspaces of the tensor product operator from scratch.