Math Problems Vault

Problem 123

If $A$ is a normal operator and if $F$ is a continuous function on $\operatorname {spec} A$ , then $\operatorname {spec} F(A) = F(\operatorname {spec} A)$ .

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Problem 124

Prove without using the spectral theorem that, for Hermitian operators, $0 \leq A \leq 1$ if and only if $A^{2} \leq A$ .

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Problem 125

If $A$ is a normal operator on a separable Hilbert space, and if $1 + A + \cdots + A^{n}$ is diagonal for some positive integer $n$ , does it follow that $A$ is diagonal?

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Remark.

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An operator is diagonal (see Problem 61) if there exists an orthonormal basis of the space consisting entirely of eigenvectors of that operator.

Problem 126

If $F$ is a continuous function on the complex plane, is the mapping $A \mapsto F(A)$ , defined for all normal operators, continuous?

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Problem 127

A bounded linear transformation $U$ is a partial isometry if and only if $U^{*}U$ is a projection.

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Remark.

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An isometry is a linear transformation $U$ (from a Hilbert space into itself, or from one Hilbert space into another) such that $\left\| Uf\right\| = \left\| f\right\|$ for all $f$ . A necessary and sufficient condition that a linear transformation $U$ be an isometry is that $U^{*}U = 1$ . Caution: the conditions $U^{*}U = 1$ and $UU^{*} = 1$ are not equivalent. The latter condition is satisfied in case $U^{*}$ is an isometry; in that case $U$ is called a co-isometry.

A partial isometry is a linear transformation that is isometric on the orthogonal complement of its kernel. It is easy to verify that a partial isometry $U$ is bounded, in fact if $U$ is not $0$ , then $\left\| U\right\| = 1$ .

The orthogonal complement of the kernel of a partial isometry is frequently called its initial space. The initial space of a partial isometry $U$ turns out to be equal to the set of all those vectors $f$ for which $\left\| Uf\right\| = \left\| f\right\|$ . The range of a partial isometry is equal to the image of the initial space and is necessarily closed. For partial isometries, the range is sometimes called the final space.

Problem 128

Discover the implication relations between the statements " $UM = M$ " and " $M$ reduces $U$ " when $U$ is a maximal partial isometry.

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Remark.

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For partial isometries $U$ and $V$ , write $U \leq V$ in case $V$ agrees with $U$ on the initial space of $U$ ; this implies that the initial space of $U$ is included in the initial space of $V$ . A partial isometry is called maximal if it is not dominated, in this order, by any partial isometry other than itself. A partial isometry is maximal if and only if either it or its adjoint is an isometry.

Problem 129

The set of all non-zero partial isometries is closed but not connected (with respect to the norm topology of operators).

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Problem 130

If $U$ and $V$ are partial isometries such that $\left\| U - V\right\| < 1$ , then $\rho (U) = \rho (V)$ , $\rho '(U) = \rho '(V)$ , and $\nu (U) = \nu (V)$ .

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Remark.

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If $U$ is a partial isometry, write $\rho (U) = \dim \operatorname {ran} U$ , $\rho '(U) = \dim \operatorname {ran}^{\perp } U$ , and $\nu (U) = \dim \ker U$ . These three cardinal numbers are called the rank, the co-rank, and the nullity of $U$ , respectively.

Problem 131

Each pair of partial isometries (on the same Hilbert space) with the same rank, co-rank, and nullity, can be joined by a continuous curve of partial isometries with the same rank, co-rank, and nullity.

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Remark.

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If $U$ is a partial isometry, write $\rho (U) = \dim \operatorname {ran} U$ , $\rho '(U) = \dim \operatorname {ran}^{\perp } U$ , and $\nu (U) = \dim \ker U$ for its rank, co-rank, and nullity, respectively.

Problem 132

If $A$ and $B$ are unitarily equivalent contractions, then $M(A)$ and $M(B)$ are unitarily equivalent, and conversely.

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Remark.

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An operator $A$ is a contraction if $\left\| A\right\| \leq 1$ ; for a contraction $A$ , the operator $1 - AA^{*}$ is positive, so there exists a unique positive operator $A'$ whose square is $1 - AA^{*}$ . The operator matrix

M(A) = \begin{pmatrix} A & A' \\ 0 & 0 \end{pmatrix}

is a partial isometry.

Problem 133

What conditions must a set of complex numbers satisfy in order that it be the spectrum of some non-unitary partial isometry?

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