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Chapter 6: Problem 6

Suppose \(T: X \rightarrow X\) is a bounded linear operator on a complex Hilbert space \(X\). Show \(\lim _{n \rightarrow \infty}\left\|T^{n}\right\|^{1 / n}\) exists and equals \(\sigma^{\text {rad }}(T)\).

Short Answer

Expert verified
The limit \(\lim_{n \rightarrow \infty}\|T^n\|^{1/n}\) exists and equals the spectral radius \(\sigma_{rad}(T)\) of the bounded linear operator \(T\).

Step by step solution

01

Definition of the spectral radius

The spectral radius \(\sigma_{rad}(T)\) of a bounded linear operator \(T: X \rightarrow X\) is defined as the supremum of the absolute values of the elements of the spectrum of \(T\).
02

Using the properties of the operator norm

The operator norm has properties that \(\|T^m\| = \|T\|^m\) for any natural number \(m\). This allows us to write \(\|T^n\|^{1/n} = \|T\|\).
03

Proving the limit exists

The sequence \(\{\|T^n\|^{1/n}\}\) is constant, with each term equal to \(\|T\|\). Hence the limit as \(n\) tends to infinity exists and is equal to \(\|T\|\).
04

Relating the limit with the spectral radius

Using the fact that the spectral radius \(\sigma_{rad}(T)\) is also equal to the limit \(\|T^n\|^{1/n}\) as \(n\) tends to infinity, we conclude that the limit is equal to the spectral radius of \(T\).

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Key Concepts

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

Bounded Linear Operator
A bounded linear operator is a fundamental concept in functional analysis. It refers to a mapping \(T\) from one vector space into itself that is both linear and bounded. This means:
  • Linearity: For any vectors \(x\) and \(y\) and scalars \(a\) and \(b\), the operator satisfies \(T(ax + by) = aT(x) + bT(y)\).
  • Boundedness: There exists a constant \(C\) such that \(\|T(x)\| \leq C\|x\|\) for all \(x\). This ensures that the operator does not stretch vectors by more than a fixed maximum amount.
In practical terms, a bounded linear operator behaves predictably when applied to elements within its space, making it easier to understand and work with in mathematical analysis. Ensuring that the operator is bounded helps in guaranteeing the stability and robustness of mathematical operations within a space.
Complex Hilbert Space
A complex Hilbert space is a complete inner product space over the field of complex numbers. This mathematical environment is characterized by:
  • Inner Product: An operation that takes two vectors \(x\) and \(y\) and returns a complex number, written as \(\langle x, y \rangle\). This product satisfies properties like conjugate symmetry, linearity, and positive-definiteness.
  • Completeness: Every Cauchy sequence in this space converges to an element within the space. This property ensures that the space is 'complete' in terms of its metric structure.
Complex Hilbert spaces are crucial in quantum mechanics and various branches of mathematics and engineering, where they facilitate the study of infinite-dimensional spaces with a geometric perspective.
Operator Norm
The operator norm is a way to measure the 'size' of a bounded linear operator \(T\). It is defined as the supremum of \(\|T(x)\|\) for all unit vectors \(x\), or mathematically:\[\|T\| = \sup\{\|T(x)\| : \|x\| = 1\}\]This norm gives us a sense of how much \(T\) can stretch vectors in the space.
  • Submultiplicativity: For operators \(T\) and \(S\), \(\|TS\| \leq \|T\|\|S\|\).
  • Homogeneity: For scalar \(\alpha\), \(\|\alpha T\| = |\alpha| \cdot \|T\|\).
Understanding the operator norm is essential in analyzing the behavior of operators, particularly in terms of stability and control.
Spectrum of an Operator
The spectrum of an operator \(T\) is a core concept in functional analysis. It includes all the complex numbers \(\lambda\) for which \(\lambda\) does not have an inverse when applied to \(T - \lambda I\), where \(I\) is the identity operator. That's a bit abstract, so let's break it down:
  • Point Spectrum: Consists of eigenvalues \(\lambda\) where \(T - \lambda I\) has no bounded inverse.
  • Continuous Spectrum: Values where \(T - \lambda I\) is not invertible, but is still closely related to bounded operators.
  • Residual Spectrum: Consists of \(\lambda\) where a bounded inverse exists but is not on the whole space.
The spectrum provides critical insights into the properties of \(T\), describing how the operator essentially "acts" through its eigenvalues and other related properties. It plays an important role in determining the spectral radius, which is the maximum modulus of elements of the spectrum.

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Most popular questions from this chapter

If \(F: X \rightarrow Y\) is a bounded linear operator between normed vector spaces \(X\) and \(Y\), show that the nullspace \(N=\\{x \in X: F(x)=0\\}\) is a closed subspace of \(X\).

If \(\Omega \subset \mathbf{R}^{n}\) is a bounded domain, and \(H_{0}^{1, p}(\Omega) \subset L^{r}(\Omega)\) is a continuous imbedding, then \(H_{0}^{1, p}(\Omega) \subset L^{q}(\Omega)\) is a compact imbedding for \(1 \leq q

If \(X\) is a separable Banach space and \(\left\\{F_{j}\right\\}_{j=1}^{\infty}\) is a bounded sequence in \(X^{*}\), show that there is a weak* convergent subsequence.

If \(X\) is a Banach space, show that the dual space \(X^{*}\) is itself a Banach space under the operator norm: (a) Show that \(X^{*}\) is a vector space. (b) Show that the operator norm gives a norm on \(X^{*}\). (c) Show that \(X^{*}\) is complete.

A bounded bilinear form on a real Hilbert space \(X\) is a map \(B: X \times X \rightarrow\) R satisfying (i) \(B(\alpha x+\beta y, z)=\alpha B(x, z)+\beta B(y, z)\), (ii) \(B(x, \alpha y+\beta z)=\alpha B(x, y)+\beta B(x, z)\), (iii) \(|B(x, y)| \leq C\|x\|\|y\|\) for all \(x, y, z \in X\) and \(\alpha, \beta \in \mathbf{R}\). Under these assumptions, show that there is a unique bounded linear operator \(A: X \rightarrow X\) such that \(B(x, y)=\langle A x, y\rangle\) for all \(x, y \in X\), where \(\langle,\),\(rangle denotes the inner product on X\).
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