Operator Zero iff Inner Product Zero
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Theorem
Let $\HH$ be a Hilbert space over $\C$.
Let $A: \HH \to \HH$ be a bounded linear operator.
Suppose that:
- $\forall h \in \HH: \innerprod {A h} h_\HH = 0$
Then $A$ is the zero operator.
Proof
For each $x, y \in \HH$ we have:
\(\ds 0\) | \(=\) | \(\ds \innerprod {\map A {x + y} } {x + y} - \innerprod {\map A {x - y} } {x - y} + i \innerprod {\map A {x + i y} } {x + i y} - i \innerprod {\map A {x - i y} } {x - i y}\) | since $\innerprod {A h} h_\HH = 0$ for each $h \in \HH$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \innerprod {A x} x + \innerprod {A x} y + \innerprod {A y} x + \innerprod {A y} y - \paren {\innerprod {A x} x - \innerprod {A x} y - \innerprod {A y} x + \innerprod {A y} y}\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds i \paren {\innerprod {A x} x + \innerprod {A x} {i y} + \innerprod {i A y} x + \innerprod {i A y} {i y} } - i \paren {\innerprod {A x} x - \innerprod {i A y} x + \innerprod {A x} {i y} - \innerprod {i A y} {i y} }\) | Inner Product is Sesquilinear | |||||||||||
\(\ds \) | \(=\) | \(\ds \innerprod {A x} y + \innerprod {A y} x - \paren {-\innerprod {A x} y - \innerprod {A y} x} + i \paren {\innerprod {A x} {i y} + \innerprod {i A y} x + \cmod i^2 \innerprod {A y} y} - i \paren {\innerprod {i A y} x + \innerprod {A x} {i y} - \cmod i^2 \innerprod {A y} y}\) | Inner Product is Sesquilinear and again using the hypothesis $\innerprod {A h} h_\HH = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \innerprod {A x} y + 2 \innerprod {A y} x + i \paren {-i \innerprod {A x} y + i \innerprod {A y} x} - i \paren {i \innerprod {A y} x - i \innerprod {A x} y}\) | using the hypothesis $\innerprod {A h} h_\HH = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \innerprod {A x} y + 2 \innerprod {A y} x + \innerprod {A x} y - \innerprod {A y} x - \innerprod {A y} x + \innerprod {A x} y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \innerprod {A x} y\) |
So:
- $\innerprod {A x} y = 0$
for each $x, y \in \HH$.
Setting $y = A x$, we have:
- $\norm {A x}^2 = 0$
by the definition of the inner product norm.
So from Norm Axiom $\text N 1$: Positive Definiteness, we have $A x = 0$ for each $x \in \HH$.
$\blacksquare$
Also see
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.15$