Math Problems Vault

Problem 44

Every operator has a column-finite matrix. More precisely, if $A$ is an operator on a Hilbert space $\mathbf{H}$ , then there exists an orthonormal basis $\left\{ e_j\right\}$ for $\mathbf{H}$ such that, for each $j$ , the matrix entry $\left\langle Ae_j, e_i \right\rangle$ vanishes for all but finitely many $i$ 's.

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Problem 45

If $\alpha_{ij} \geq 0$ , if $p_i > 0$ and $q_j > 0$ $(i, j = 0, 1, 2, \ldots )$ , and if $\beta$ and $\gamma$ are positive numbers such that

\begin{align} \sum _i \alpha _{ij} p_i & \leq \beta q_j & (j = 0, 1, 2, \ldots ), \\ \sum _j \alpha _{ij} q_j & \leq \gamma p_i & (i = 0, 1, 2, \ldots ), \end{align}

then there exists an operator $A$ (on a separable infinite-dimensional Hilbert space, of course) with $\left\| A\right\|^2 \leq \beta \gamma$ and with matrix $\left\langle \alpha_{ij} \right\rangle$ (with respect to a suitable orthonormal basis).

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Problem 46

There exists an operator $A$ (on a separable infinite-dimensional Hilbert space) with $\left\| A\right\| \leq \pi$ and with matrix $\left\langle 1/(i+j+1) \right\rangle$ $(i, j = 0, 1, 2, \ldots )$ .

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Remark.

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This matrix is named after Hilbert and is accordingly called the Hilbert matrix; the norm of the matrix is in fact equal to $\pi$ .

Problem 47

The matrix $\left\langle 2^{-(i+j+1)} \right\rangle$ is bounded. What is its norm?

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Remark.

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A matrix whose $\left\langle i, j \right\rangle$ entry is a function of $i+j$ only is called a Hankel matrix. The Hilbert matrix (Problem 46) is the Hankel matrix that corresponds to the function $\varphi$ defined by $\varphi (x) = 1/(x+1)$ (i.e., $\alpha_{ij} = \varphi (i+j)$ , $i, j = 0, 1, 2, \ldots$ ); the matrix $\left\langle 2^{-(i+j+1)} \right\rangle$ is the Hankel matrix corresponding to $\varphi (x) = 2^{-(x+1)}$ .

Problem 48

Is the Hilbert matrix positive?

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Remark.

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The Gramian of a finite or infinite sequence $\left\{ f_n\right\}$ of vectors is the matrix whose $\left\langle i, j \right\rangle$ entry is the inner product $\left\langle f_i, f_j \right\rangle$ .

Problem 49

Under what conditions on a sequence $\left\{ f_n\right\}$ of vectors in a Hilbert space $\mathbf{H}$ does the series $\sum_n \alpha_n f_n$ converge in $\mathbf{H}$ for every sequence $\alpha$ in $l^2$ ?

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