14.4.2 Tensor Linear Operator Product Domain Space

Tensor Linear Operator Product Domain Space is the tensor product V tensor W of the two spaces on which the individual operators f and g act, serving as the single source space of the induced operator f tensor g and, since the operator case forces domain and codomain to coincide, doubling simultaneously as its codomain.

Formation of the Domain Space

Building the Space from the Two Operator Domains

Given operators

the domain space of the induced operator is formed as

built from V and W exactly as in the general tensor product construction, with no special modification required by the fact that f and g happen to be operators rather than maps between unrelated spaces.

Dimension of the Domain Space

When V and W are finite-dimensional with dimensions m and n respectively, the domain space V tensor W has dimension m n, and this dimension is what determines the size of the square matrix representing f tensor g, namely an m n by m n matrix, once bases of V and W are fixed.

Elements of the Domain Space

Elementary Tensors as Generators

The domain space is spanned by elementary tensors v tensor w with v in V and w in W, and every element of the domain space, though not necessarily an elementary tensor itself, is expressible as a finite sum of such elementary tensors, providing the generating set on which the action of f tensor g is directly specified before being extended by linearity.

Basis Construction

If e-1 through e-m is a basis of V and h-1 through h-n is a basis of W, then the m n elementary tensors e-i tensor h-j form a basis of the domain space, and this induced basis is precisely the one used to express f tensor g as the Kronecker product of the matrices of f and g.

Role as Both Domain and Codomain

Coincidence Forced by the Operator Case

Because f maps V to itself and g maps W to itself, the factor space relation assigns V tensor W as both the domain and the codomain of f tensor g, a coincidence that does not occur in the general tensor product of maps construction, where domain and codomain tensor spaces are typically built from four distinct spaces.

Consequence for Iterating the Construction

Because the domain space doubles as the codomain space, the induced operator f tensor g can be composed with itself, or tensored again with a further pair of operators acting on further copies of the domain space, without any adjustment to the ambient space, a convenience specific to the operator case that is not automatically available when domain and codomain differ.

Interaction with Subspaces of the Domain Space

Invariant Subspaces Built from Factor Subspaces

If U is a subspace of V invariant under f, and X is a subspace of W invariant under g, then U tensor X is a subspace of the domain space V tensor W invariant under f tensor g, giving a systematic source of invariant subspaces of the domain space directly from invariant subspaces of the two factors.

Restriction to a Product Subspace

Restricting f tensor g to a subspace of the domain space of the form U tensor X, where U and X are invariant under f and g respectively, produces an operator equal to the tensor product of the restrictions of f to U and of g to X, so the domain space of this restricted operator is the smaller space U tensor X rather than the full V tensor W, illustrating how the domain space of the construction shrinks consistently when passing to invariant product subspaces.