Tensor Products of Linear Operators of Normed Linear Spaces

Definition: Let $X$, $Y$, $W$, and $Z$ be normed linear spaces and let $S : X \to W$ and $T : Y \to Z$ be linear operators. Define the Tensor Product of the Linear Operators $S$ and $T$ which we will denote by $S \otimes T : X \otimes Y \to W \otimes Z$ to be the unique linear map with the property that $(S \otimes T)(x \otimes y) = S(x) \otimes T(y)$ for all $x \in X$ and $y \in Y$.

Note that if $S : X \to W$ and $T : Y \to Z$ are linear operators then let $A : X \times Y \to W \otimes Z$ be defined for all $x \in X$ and $y \in Y$ by:

(1)

\begin{align} \quad A(x, y) = S(x) \otimes T(y) \end{align}

It is clear that $A : X \times Y \to W \otimes Z$ is bilinear since:

(2)

\begin{align} \quad A(x_1 + x_2, y) &= S(x_1 + x_2) \otimes T(y) = [S(x_1) + S(x_2)] \otimes T(y) = S(x_1) \otimes T(y) + S(x_2) \otimes T(y) = A(x_1, y) + A(x_2, y) & \forall x_1, x_2 \in X, \: \forall y \in Y \\ \quad A(\alpha x, y) &= S(\alpha x) \otimes T(y) = [\alpha S(x)] \otimes T(y) = \alpha [S(x) \otimes T(y)] = \alpha A(x, y) & \forall x \in X, \: \forall y \in Y, \: \forall \alpha \in \mathbf{F} \end{align}

(3)

\begin{align} \quad A(x, y_1 + y_2) &= S(x) \otimes T(y_1 + y_2) = S(x) \otimes [T(y_1) + T(y_2)] = S(x) \otimes T(y_1) + S(x) \otimes T(y_2) = A(x, y_1) + A(x, y_2) & \forall x \in X, \: \forall y_1, y_2 \in Y \\ \quad A(x, \alpha y) &= S(x) \otimes T(\alpha y) = S(x) \otimes [\alpha T(y)] = \alpha [S(x) \otimes T(y)] = \alpha A(x, y) & \forall x \in X, \: \forall y \in Y, \: \forall \alpha \in \mathbf{F} \end{align}

So by the theorem on The Existence of a Linear Map σ on X⊗Y to Z that Matches a Bilinear Map on X×Y to Z page there exists a unique linear map $\tilde{A} : X \otimes Y \to W \otimes Z$ such that $\tilde{A}(x \otimes y) = A(x, y)$ for all $x \in X$ and for all $y \in Y$. We use the notation $S \otimes T$ instead of $\tilde{A}$ and we have that:

(4)

\begin{align} \quad (S \otimes T)(x \otimes y) = A(x, y) = S(x) \otimes T(y) \quad \forall x \in X, \: \forall y \in Y \end{align}

Proposition 1: Let $X$, $Y$, $Z$, and $W$ be normed linear spaces and let $S : X \to W$, $T : Y \to Z$ be linear operators. Let $S \otimes T : X \otimes Y \to W \otimes Z$ be the tensor product of the operators $S$ and $T$ as defined above. Then: a) If $S$ and $T$ are both injective then $S \otimes T$ is injective. b) If $S$ and $T$ are both surjective then $S \otimes T$ is surjective. c) If $S$ and $T$ are both bijective then $S \otimes T$ is bijective.

The Projective Tensor Product of Linear Operators

Proposition 2: Let $X$, $Y$, $W$, and $Z$ be normed linear spaces and let $S : X \to W$ and $T : Y \to Z$ be linear operators. Then there exists a unique linear operator which we denote by $S \otimes_p T : X \otimes_p Y \to W \otimes_p Z$ with the following properties: 1) $(S \otimes_p T)(x \otimes y) = S(x) \otimes T(y)$ for all $x \in X$ and for all $y \in Y$. 2) $\| S \otimes_p T \| = \| S \| \| T \|$.

Definition: Let $X$, $Y$, $W$, and $Z$ be normed linear spaces and let $S : X \to W$ and $T : Y \to Z$ be linear operators. Define the Projective Tensor Product of the Linear Operators $S$ and $T$ which we will denote by $S \otimes_p T : X \otimes_p Y \to W \otimes_p Z$ to be the unique linear map with the property that $(S \otimes T)(x \otimes y) = S(x) \otimes T(y)$ for all $x \in X$ and $y \in Y$

The Weak/Injective Tensor Product of Linear Operators

Proposition 3: Let $X$, $Y$, $W$, and $Z$ be normed linear spaces and let $S : X \to W$ and $T : Y \to Z$ be linear operators. Then there exists a unique linear operator which we denote by $S \otimes_w T : X \otimes_w Y \to W \otimes_w Z$ with the following properties: 1) $(S \otimes_w T)(x \otimes y) = S(x) \otimes T(y)$ for all $x \in X$ and for all $y \in Y$. 2) $\| S \otimes_w T \| = \| S \| \| T \|$.

Definition: Let $X$, $Y$, $W$, and $Z$ be normed linear spaces and let $S : X \to W$ and $T : Y \to Z$ be linear operators. Define the Weak/Injective Tensor Product of the Linear Operators $S$ and $T$ which we will denote by $S \otimes_w T : X \otimes_w Y \to W \otimes_w Z$ to be the unique linear map with the property that $(S \otimes T)(x \otimes y) = S(x) \otimes T(y)$ for all $x \in X$ and $y \in Y$