Two definitions of spectrums

Question

In Kreyszig's Introductory Functional Analysis Page 371, the point spectrum is defined as $\sigma_p(T)$ such that $R_\lambda(T) = (T - \lambda I)^{-1}$ does not exist.

While in my functional analysis class lecture notes, the point spectrum is defined as $\sigma_p(T) = \{\lambda \in \mathbb C : ker(T - \lambda I) \neq \{0\}\}.$ The spectrum is defined as $\sigma(T) = \{T_\lambda := T - \lambda I$ is not invertible$\}.$

It seems to me that such two definitions are contradictory, that Kreyszig's definition of point spectrum is the definition of spectrum in my class lecture notes. Can someone explain to me the difference of such definitions?

Answer 1

The difference is due to the fact that Kreyszig consider the spectrum of unbounded operators. So for $\lambda$ in the continuous or residual spectrum, $R_\lambda$ exists (as an unbounded operator).

The usual approach in functional analysis is for bounded operators, where you wouldn't allow for $R_\lambda$ to be unbounded, and so you would say it doesn't exist.

Answer 2

Usually the point spectrum consists only of the eigenvalues of $T$, that is, $$\sigma_p(T) = \{\lambda\in\mathbb{C}\ |\ T-\lambda I\text{ is not injective }\}.$$

Notice how this corresponds to the eigenvalues of $T$, since when $T-\lambda I$ is not injective, there exists some $x\neq 0$, such that $(T-\lambda I)x=0$, or $Tx=\lambda x$. My definition of the point spectrum is equivalent to the one you gave, namely

$$\sigma_p(T) = \{\lambda\in\mathbb{C}\ |\ \ker{(T-\lambda I)\neq 0}\}.$$

The definition of the spectrum is usually

$$\sigma(T) = \{\lambda\in\mathbb{C}\ |\ T-\lambda I\text{ is not invertible}\}.$$

Certainly $\sigma_p(T)\subseteq\sigma(T)$, but on infinite-dimensional spaces, you may have that the operator $T-\lambda I$ is injective, but not surjective: Some $\lambda$ may not be an eigenvalue of $T$, but still the operator $T-\lambda I$ is not invertible.