Math Problems Vault

Problem 147 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 17 "Unilateral Shift"

The commutant of the unilateral shift is the set of all restrictions to $\mathbf{H}^2$ of multiplications by multipliers in $\mathbf{H}^{\infty }$ .

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Remark.

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The commutant of an operator (or of a set of operators) is the set of all operators that commute with it (or with each operator in the set). The unilateral shift is the restriction of the bilateral shift $W$ to $\mathbf{H}^2$ , where $\mathbf{H}^2$ is the closed span in $L^2$ of the unit circle of the functions $e_n$ , $n = 0, 1, 2, \ldots$ (with $e_n(z) = z^n$ ), and $\mathbf{H}^{\infty }$ is the set of essentially bounded functions in $\mathbf{H}^2$ . If the bilateral shift is regarded as a multiplication (by $e_1$ on $L^2$ ), then its commutant is the set of all multiplications on the same $L^2$ (Problem 146). If, however, the multiplier itself is in $\mathbf{H}^2$ (and hence in $\mathbf{H}^{\infty }$ ), then $\mathbf{H}^2$ is invariant under the induced multiplication (cf. Problem 34), and the restriction of that multiplication to $\mathbf{H}^2$ makes sense.

Problem 155 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 17 "Unilateral Shift"

If $U$ is the (simple) unilateral shift and if $\mathbf{M}$ is a non-zero subspace invariant under $U$ , then there exists a (necessarily unique) one-dimensional wandering subspace $\mathbf{N}$ such that $\mathbf{M} = \bigvee_{n=0}^{\infty } U^n \mathbf{N}$ .

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Remark.

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If $A$ is an operator on a Hilbert space $\mathbf{H}$ , a subspace $\mathbf{N}$ of $\mathbf{H}$ is called wandering for $A$ if it is orthogonal to all its images under the positive powers of $A$ .

Problem 157 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 17 "Unilateral Shift"

A non-zero subspace $\mathbf{M}$ of $\mathbf{H}^2$ is invariant under $U$ if and only if there exists a function $\varphi$ in $\mathbf{H}^{\infty }$ , of constant modulus $1$ almost everywhere, such that $\mathbf{M}$ is the range of the restriction to $\mathbf{H}^2$ of the multiplication induced by $\varphi$ .

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Remark.

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Regard $U$ as the restriction to $\mathbf{H}^2$ of the multiplication induced by $e_1$ on $L^2$ of the unit circle. A function $\varphi$ in $\mathbf{H}^{\infty }$ of constant modulus $1$ almost everywhere is called an inner function; informally, $\mathbf{M} = \varphi \cdot \mathbf{H}^2$ can be described as the set of all multiples of $\varphi$ by functions in $\mathbf{H}^2$ .

Problem 2.17.23 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

Find the conditions mentioned in Sec. 17.22.

Section 17.22 (Remark): Let $S$ be the shift operator on $H^{2}$ , $(S f)(z) = z f(z)$ $(f \in H^{2}, z \in U)$ ; a closed subspace $Y$ of $H^{2}$ with $S(Y) \subset Y$ is called $S$ -invariant. By Beurling's theorem (Theorem 17.21), every closed $S$ -invariant subspace of $H^{2}$ other than $\left\{ 0\right\}$ is of the form $\varphi H^{2} = \left\{ \varphi f: f \in H^{2}\right\}$ for an inner function $\varphi$ , and $\varphi$ is determined by the subspace up to a constant factor; by Theorem 17.15, every inner function has the form

M(z) = c B(z) \exp \left\{ -\int _{-\pi }^{\pi } \frac{e^{i t}+z}{e^{i t}-z} d \mu (t)\right\} \quad (z \in U),

where $c$ is a constant, $\left|c\right| = 1$ , $B$ is a Blaschke product, and $\mu$ is a finite positive Borel measure on $T$ which is singular with respect to Lebesgue measure. Combining these, the $S$ -invariant subspaces of $H^{2}$ are characterized by the following data: a sequence of complex numbers $\left\{ \alpha_{n}\right\}$ (possibly finite, or even empty) such that $\left|\alpha_{n}\right| < 1$ and $\Sigma \left(1-\left|\alpha_{n}\right|\right) < \infty$ (the zeros of $B$ ), and a positive Borel measure $\mu$ on $T$ , singular with respect to Lebesgue measure. The book states that it is easy to find conditions, in terms of $\left\{ \alpha_{n}\right\}$ and $\mu$ , which ensure that one $S$ -invariant subspace of $H^{2}$ contains another.

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Beurling's theorem on shift-invariant subspaces

References