8.2 Spectral mapping theorem and functional calculus

The spectral mapping theorem connects an operator's spectrum to functions applied to it. It's a powerful tool for understanding how transformations affect an operator's properties, allowing us to analyze complex operators through simpler functions.

Functional calculus extends this idea, letting us apply continuous or even measurable functions to operators. This broadens our toolkit for studying operators, enabling us to construct and analyze more sophisticated mathematical objects in functional analysis.

Spectral Mapping Theorem

Spectral mapping theorem

Let $T$ be a bounded linear operator on a Banach space $X$ and $f$ be a complex-valued function analytic on an open set containing $\sigma(T)$, the spectrum of $T$. The spectral mapping theorem states that $\sigma(f(T)) = f(\sigma(T))$, meaning the spectrum of $f(T)$ is the image of the spectrum of $T$ under $f$. The proof involves showing that if $\lambda \in \sigma(T)$ and $z = f(\lambda)$, then $f(T) - zI$ is not invertible, implying $z \in \sigma(f(T))$. Conversely, if $z \in \sigma(f(T))$, there exists $\lambda \in \sigma(T)$ such that $f(\lambda) = z$, proving the equality $f(\sigma(T)) = \sigma(f(T))$

Applications of spectral mapping

To find the spectrum of $f(T)$, where $T$ is a bounded linear operator and $f$ is an analytic function:

1. Compute the spectrum of $T$, denoted by $\sigma(T)$
2. Apply the function $f$ to each element of $\sigma(T)$
3. The resulting set is the spectrum of $f(T)$, denoted by $\sigma(f(T))$

For example, let $T$ be a bounded linear operator with $\sigma(T) = \{1, 2, 3\}$ and $f(z) = z^2$. Applying $f$ to each element of $\sigma(T)$ yields $f(1) = 1$, $f(2) = 4$, and $f(3) = 9$. Therefore, $\sigma(f(T)) = \sigma(T^2) = \{1, 4, 9\}$

Functional Calculus

Continuous functional calculus

The continuous functional calculus extends the spectral mapping theorem to continuous functions. Let $T$ be a bounded linear operator on a Banach space $X$ with spectrum $\sigma(T)$ and $f$ be a continuous function on $\sigma(T)$. The continuous functional calculus defines an operator $f(T)$ as follows:

• If $\sigma(T)$ is real, then $f(T) = \int_{\sigma(T)} f(\lambda) dE(\lambda)$, where $E$ is the spectral measure associated with $T$
• If $\sigma(T)$ is complex, then $f(T) = \int_{\sigma(T)} f(z) dE(z)$, where the integral is taken over the complex plane

Properties of the continuous functional calculus include:

• If $f$ and $g$ are continuous functions on $\sigma(T)$, then $(f + g)(T) = f(T) + g(T)$ and $(fg)(T) = f(T)g(T)$
• If $f_n$ is a sequence of continuous functions converging uniformly to $f$ on $\sigma(T)$, then $f_n(T)$ converges to $f(T)$ in the operator norm

Borel functional calculus

The Borel functional calculus extends the functional calculus to include unbounded Borel measurable functions. Let $T$ be a bounded linear operator on a Banach space $X$ with spectrum $\sigma(T)$ and $f$ be a Borel measurable function on $\sigma(T)$. The Borel functional calculus defines an operator $f(T)$ as $f(T) = \int_{\sigma(T)} f(\lambda) dE(\lambda)$, where $E$ is the spectral measure associated with $T$. The integral is interpreted as a Bochner integral, which extends the Lebesgue integral to Banach space-valued functions.

Properties of the Borel functional calculus include:

• If $f$ and $g$ are Borel measurable functions on $\sigma(T)$, then $(f + g)(T) \subseteq f(T) + g(T)$ and $(fg)(T) \subseteq f(T)g(T)$
• If $f_n$ is a sequence of Borel measurable functions converging pointwise to $f$ on $\sigma(T)$, then $f_n(T)$ converges strongly to $f(T)$

The Borel functional calculus allows for the construction of operators such as the square root ($\sqrt{T}$), logarithm ($\log T$), and exponential ($e^T$) of a bounded linear operator