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Chapter 13: Problem 13

If \(A: \mathcal{H} \rightarrow \mathcal{H}\) is an operator such that \(A u \perp u\) for all \(u \in \mathcal{H}\), show that \(A=0\).

Short Answer

Expert verified
The operator \(A\) is the zero operator (\(A = 0\)) as it maps all vectors in the space \(\mathcal{H}\) to the zero vector.

Step by step solution

01

Recall Properties of Orthogonality and Dot Product

We must first remember the principle that if two vectors are orthogonal, their dot product is zero. This is denoted as \(u \cdot v = 0\) for \(u \perp v\). It follows that \(u \cdot A u = 0\) for all \(u \in \mathcal{H}\), under the condition \(A u \perp u\).
02

Expand the Dot Product

Now, expand the dot product using the linearity property of the dot product and distribute it over the operator \(A\): \(u \cdot A u = \| A u \|^2 = 0\). Since the square norm of a vector is zero, the vector itself must be the zero vector. This implies that the vector \(Au\) is the zero vector for every vector \(u\) in the space \(\mathcal{H}\).
03

Conclude Proof

We have shown that the resulting vector \(Au\) is a zero vector for any given vector \(u\). Consequently, the operator \(A\) must be the zero operator, denoted as \(A=0\), because it maps every vector in the space \(\mathcal{H}\) to zero.

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Most popular questions from this chapter

An operator \(A\) is called nermal if it is bounded and commutes with its adjoint. \(A^{*} A=A A^{*} .\) Show that the operator $$ A \psi(x)=c \psi(x)+l \int_{a}^{t} K(x, y) \psi(y) \mathrm{d} y $$ on \(L^{2}([a, b])\), where \(c\) is a real number and \(K(x, y)=\overline{K(y, x)}\), is normal. (a) Show that an operator \(A\) is normal if and only if \(\|A u\|=\left\|A^{*} u\right\|\) for all vectors \(u \in \mathcal{H}\). (b) Show that if \(A\) and \(B\) are commuting normal operators, \(A B\) and \(A+\lambda B\) are normal for all \(\lambda \in \mathbb{C}\)

If \(A\) is a symmetric operator, show that \(A^{*}\) is symmetric if and only if it is self-edjoint, \(A^{*}=A^{* *}\)

Show that a non-zero vector \(u\) is an cigenvector of an operator \(A\) if and only if \(|\langle u \mid A u\rangle|=\| A u|| u \mid .\)

If \(A\) is a sclf-adjoint operator show that $$ \|(A+t) u\|^{2}=\|A u\|^{2}+I u \|^{2} $$ and that the operator \(A+1 I\) is invertible, Show that the operator \(U=\left(A-{ }_{1} I\right)(A+i I)^{-1}\) is unitary (called the Cayley transform of \(A\) ).

Show that a vector subspace is a closed subset of \(\mathcal{H}\) with respect to the norm topology iff the limit of every sequence of vectors in \(V\) belongs to \(V\).
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