Woo Young Lee Lecture Notes on Operator Theory

Chapter 2
Weyl Theory
2.1 Introduction
In 1909, writing about differential equations, Hermann Weyl noticed something about
the essential spectrum of a self adjoint operator on Hilbert space: when you take it
away from the spectrum, you are left with the isolated eigenvalues of finite multiplicity.
This was soon generalized to normal operators, and then to more and more classes of
operators, bounded and unbounded, on Hilbert and on Banach spaces.
The spectrum σ(T ) of a bounded linear operator T on a complex Banach space
X is of course the set of those complex numbers for which T − λI does not have an
everywhere defined two-sided inverse: this concept extends at once to the spectrum
σ A (a) of a Banach algebra element a ∈ A. Thus the Fredholm essential spectrum
σ e (T ) is the spectrum of the coset T + K(X) of the operator T ∈ B(X) in the Calkin
algebra B(X)/K(X). Equivalently λ ∈ C is excluded from the spectrum σ(T ) if
and only if operator T − λI is one one and onto, and is excluded from the essential
spectrum σ e (T ) if and only if the operator T − λI has finite dimensional null space
and range of finite co dimension.
The Fredholm essential spectrum is contained in the larger Weyl spectrum, which
also includes points λ ∈ C for which T − λI is Fredholm but with non zero index: the
two finite dimensions involved are unequal. Equivalently, T − λI /∈ B(X) −1 + K(X)
cannot be expressed as the sum of an invertible and a compact operator. What is
relevant here is that for self adjoint and more general normal operators the Weyl
and the Fredholm spectra coincide: every normal Fredholm operator has index zero.
Thus while the original Weyl observation of 1909 may have seemed to subtract the
Fredholm essential spectrum from the spectrum, it can equally be interpreted as
subtracting the Weyl essential spectrum. For non normal operators it is this modified
version that seems to be the property that is of interest. For a linear operator on
a Banach space the most obvious points of its spectrum are the eigenvalues π 0 (T ),
collecting λ ∈ C for which T − λI fails to be one-one. As is familiar from matrix
theory, in finite dimensions this is all of the spectrum. In a sense therefore Weyl’s
theorem seems to be suggesting that for nice operators the spectrum splits into a
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