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Chapter 3: Problem 5

(Weighted shifts.) Given a bounded sequence \(\left(\lambda_{n}\right)\) in \(\mathbb{C}\) define an operator \(S\) in \(\mathbf{B}\left(\ell^{2}\right)\) by \((S x)_{1}=0\) and $$ (S x)_{n}=\lambda_{n} x_{n-1}, \quad n>1, \quad \text { for } x=\left(x_{n}\right) \text { in } \ell^{2} . $$ Find the polar decomposition of \(S\), and characterize those sequences \(\left(\lambda_{n}\right)\) in \(\ell^{\infty}\) for which \(S\) is compact.

Short Answer

Expert verified
The operator \( S \) is compact if \( \lambda_n \to 0 \) as \( n \to \infty \).

Step by step solution

01

Understand the Operator S

The operator \( S \) is defined on the Hilbert space \( \ell^2 \). For any sequence \( x = (x_n) \) in \( \ell^2 \), which is a space of square-summable sequences, the operator acts as \((S x)_1 = 0\) and \((S x)_n = \lambda_n x_{n-1}\) for \( n > 1\). This definition implies that \( S \) shifts each element of the sequence by one position to the right, and multiplies it by the weight \( \lambda_n \).
02

Polar Decomposition of S

The polar decomposition of an operator \( S \) is \( S = U |S| \), where \( |S| = (S^*S)^{1/2} \), and \( U \) is a partial isometry. First, compute \( S^* \): For \((y_n) \in \ell^2\), we need \( \langle Sx, y \rangle = \langle x, S^*y \rangle \). This gives \( (S^*y)_n = \lambda_{n+1} \overline{y}_{n+1} \) for \( n \geq 1 \) and \( (S^*y)_n = 0 \) for \( n < 1 \). Then, \( S^*S = \text{diag}(|\lambda_2|^2, |\lambda_3|^2, |\lambda_4|^2, \ldots) \). Thus, \( |S| = (S^*S)^{1/2} \) is \( \text{diag}(|\lambda_2|, |\lambda_3|, \ldots) \). Finally, \( U \) is defined on \( \text{range}(|S|) \) by aligning directions of vectors appropriately.
03

Conditions for S to be Compact

An operator \( S \) is compact if for every bounded sequence \((x_n) \subset \ell^2\), the sequence \((Sx_n)\) has a converging subsequence. Since \( S \) has components \( \lambda_n x_{n-1} \), \( S \) is compact if \( \lambda_n \) converges to zero as \( n \to \infty \). Hence, the sequence \( (\lambda_n) \) must tend to zero for \( S \) to be compact. This is a consequence of the fact that the finite-rank operators are compact and a shift operator like \( S \) is composed of terms that center around these bounds when the \( \,\lambda_n \) tend to zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hilbert Space
In functional analysis, a Hilbert space is a complete inner product space, which is a vector space equipped with an inner product. This inner product allows for the measurement of angles and lengths, similar to how we measure these in Euclidean spaces. A few core properties of Hilbert spaces include:
  • The concept of orthogonality: Two elements in a Hilbert space can be orthogonal if their inner product is zero.
  • Completeness: Every Cauchy sequence of elements in a Hilbert space converges to an element within the space.
  • Infinite dimensionality: Hilbert spaces can be infinite-dimensional, allowing for a broader analysis of sequences and functions.
In the context of the given exercise, the Hilbert space is \( \ell^2 \), which is comprised of all square-summable sequences. This space is significant due to its "lattice-like" structure, providing a framework for analyzing operators like \( S \), the weighted shift operator.
Operator Theory
Operator theory is a branch of mathematics dealing with the study of operators, which are a generalization of functions, on spaces. Particularly, these operators often act on vector spaces, and in the context of functional analysis, they act on Hilbert spaces.
  • Bounded Operators: These are operators where there exists a constant \( C \) such that \( \| Sx \| \leq C \| x \| \) for all \( x \). This makes them crucial for ensuring stability in calculations.
  • Polar decomposition: A mathematical method that breaks down an operator \( S \) into two parts: a partial isometry \( U \) and a positive operator \( |S| = (S^*S)^{1/2} \).
Understanding these concepts helps in analyzing the behavior of the operator \( S \), as we break its functionality into manageable parts and comprehend its impact on sequences within the Hilbert space \( \ell^2 \).
Compact Operators
Compact operators are to functional analysis what finite matrices are to linear algebra 鈥� they transform infinite-dimensional problems into finite approximations. These operators map bounded sets to relatively compact sets, meaning forced convergence.A key property of compact operators is that they can approximate operators through finite-rank operators. This means any bounded sequence \((x_n)\) in the domain has a subsequence for which \((Sx_n)\) converges.
  • In terms of weighted shift operators like \( S \), compactness occurs if and only if the weights \( \lambda_n \) tend to zero as \( n \to \infty \).
  • The significance of this behavior is that we can effectively "compress" the operator's effect, respecting convergence within the space \( \ell^2 \).
This property makes compact operators powerful in practically simulating real-world finite conditions through an infinite-dimensional lens, efficiently simplifying analysis by focusing on these operators.
Shift Operators
Shift operators take elements of a sequence and "shift" them by a certain number of positions. In our exercise, the operator \( S \) is a weighted shift operator acting on \( \ell^2 \), defined by shifting to the right while multiplying each element of the sequence by a weight \( \lambda_n \).This type of operator can be broken down as follows:
  • Right shift: Moves each element of the sequence to the subsequent position in the sequence, filling the initial position with a zero.
  • Weighted: Each moved element is multiplied by a corresponding weight, \( \lambda_n \), before placing it in its new position.
Understanding the behavior of shift operators is essential. They are a mathematical tool reflecting various physical systems, represented through signal processing or time series analysis. Studying their properties, such as in the polar decomposition or characterizing compactness, allows us to see how operators like \( S \) interact and manipulate sequences in ways that offer meaningful insights, especially about convergence and stability in mathematical and real-world applications.

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Most popular questions from this chapter

Let \(\left\\{e_{n} \mid n \in \mathbb{N}\right\\}\) be an orthonormal basis for the Hilbert space \(\mathfrak{y}\), and define for each \(T\) in \(\mathbf{B}(\mathfrak{)})\) the doubly infinite matrix \(A=\left(\alpha_{n m}\right)\), by letting \(\alpha_{n m}=\left(T e_{m} \mid e_{n}\right) .\) Show that every row and every column in \(A\) is square summable (i.e. belongs to \(\ell^{2}\) ). Use this to prove that the matrix product \(A B=C, C=\left(\gamma_{n m}\right)\), where \(\gamma_{n m}=\sum_{k} \alpha_{n k} \beta_{k m}\), is welldefined when the matrices \(A\) and \(B\) correspond to operators \(T\) and \(S\) in \(\mathbf{B}(\mathfrak{)})\). Show that \(C\) is the matrix corresponding to the operator \(T S\). Also find the matrices corresponding to the operators \(S+T\) and \(T^{*}\).

(Volterra operators.) Let \(I=[0,1]\) and consider functions \(k\) in \(L^{2}\left(I^{2}\right)\) such that \(k(x, y)=0\) for \(x

(Fourier transform.) In this and the next two exercises we use the normalized Lebesgue integral on \(\mathbb{R}\) [i.e. the usual one divided by \(\left.(2 \pi)^{1 / 2}\right]\). One consequence of this is that \(\int g_{0}(x) d x=1\), where \(g_{0}(x)=\) \(\exp \left(-\frac{1}{2} x^{2}\right) .\) We shall use the abbreviated notation \(L^{p}\) for \(L^{p}(\mathbb{R})\). For each \(f\) in \(L^{1}\) define \(f(y)=\int f(x) \exp (-\mathrm{i} x y) d x\). (a) Show that if \(f \in L^{1}\) and id \(\cdot f \in L^{1}[\) where \(i d(x)=x]\), then \(f\) is a differentiable function and \(f^{\prime}(y)=-\mathrm{i}(i d \cdot f)^{\wedge}(y)\). (b) Show that if \(f\) is differentiable and both \(f\) and \(f^{\prime}\) belong to \(L^{1}\), then \(\left(f^{\prime}\right)^{\wedge}(y)=\mathrm{i} y f(y)\). (c) Let \(\mathscr{S}\) (the Schwartz space) denote the space of \(C^{\infty}\)-functions on \(\mathbb{R}\) such that \((i d)^{n} f^{(m)} \in L^{1}\) for all \(n\) and \(m\). Use (a) and (b) to show that \(f \in \mathscr{S}\) for every \(f\) in \(\mathscr{S}\). (d) Show that the map \(F: f \rightarrow f\) belongs to \(\mathbf{B}\left(L^{1}, C_{0}(\mathbb{R})\right)\). Hints: (a) Use Lebesgue's theorem on majorized convergence (6.1.15) on the difference quotient of \(f .\) (b) Use integration by parts and the fact that \(f \in C_{0}(\mathbb{R})\). (d) For any polynomial \(p\) we have \(p g_{0} \in \mathscr{S}\). Use this to show that \(\mathscr{S}\) is dense in \(L^{1}\).

Let \(T\) denote the unit circle in \(\mathbb{C}\) and consider the Hilbert space \(L^{2}(T)\) (with respect to Lebesgue measure on the circle). Show that the functions $$ e_{n}(z)=(2 \pi)^{-1 / 2} z^{n}, \quad n \in \mathbb{Z}, $$ form an orthonormal basis for \(L^{2}(T)\).

(The Riemann-Lebesgue lemma.) Consider a function \(f\) in \(L^{1}([0,2 \pi])\) with Fourier coefficients $$ f(n)=(2 \pi)^{-1} \int_{0}^{2 \pi} f(x) \exp (-\mathrm{i} n x) d x . $$ Show that \(f(n) \rightarrow 0\) as \(n \rightarrow \pm \infty\).
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