Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
theorem due to Apostol, Foias and Voiculescu, is equivalent to the fact that T is
quasitriangular (cf. [Pe, Theorem 1.31]). Evidently, every commuting n-tuple of
quasitriangular operators has the quasitriangular property. Also if a commuting n-
tuple T = (T 1 , · · · , T n ) has a coordinate whose adjoint has no eigenvalues then T has
the quasitriangular property.
As we have seen in the above, the inclusion (2.44) cannot be reversible even though
T = (T 1 , · · · , T n ) is a doubly commuting n-tuple (i.e., [T i , T ∗ j ] ≡ T iT ∗ j − T ∗ j T i = 0
for all i ≠ j) of hyponormal operators. On the other hand, R. Curto [Cu1, Corollary
3.8] showed that if T = (T 1 , · · · , T n ) is a doubly commuting n-tuple of hyponormal
operators then
T is Taylor invertible [Taylor Fredholm] ⇐⇒
n∑
T i Ti
∗
i=1
is invertible [Fredholm].
(2.47)
On the other hand, many authors have considered the joint version of the Browder
spectrum. We recall ([BDW], [CuD], [Da1], [Da2], [Har4], [JeL], [Sn]) that a commuting
n-tuple T = (T 1 , · · · , T n ) is called Taylor Browder if T is Taylor Fredholm
and there exists a deleted open neighborhood N 0 of 0 ∈ C n such that T − λ is Taylor
invertible for all λ ∈ N 0 . The Taylor Browder spectrum, σ Tb (T ), is defined by
σ Tb (T ) = {λ ∈ C n : T − λ is not Taylor Browder}.
Note that σ Tb (T ) = σ Te (T ) ∪ acc σ T (T ), where acc(·) denotes the set of accumulation
points. We can easily show that
σ Tw (T ) ⊂ σ Tb (T ). (2.48)
Indeed, if λ /∈ σ Tb (T ) then T − λ is Taylor Fredholm and there there exists δ > 0 such
that T − λ − µ is Taylor invertible for 0 < |µ| < δ. Since the index is continuous it
follows that index(T − λ) = 0, which says that λ /∈ σ Tw (T ), giving (2.48).
If T = (T 1 , · · · , T n ) is a commuting n-tuple, we write π 00 (T ) for the set of all
isolated points of σ T (T ) which are joint eigenvalues of finite multiplicity and write
R(T ) ≡ iso σ T (T ) \ σ Te (T ) for the Riesz points of σ T (T ). By the continuity of the
index, we can see that R(T ) = iso σ T (T ) \ σ Tw (T ).
Lemma 2.5.3. If T = (T 1 , · · · , T n ) is a commuting n-tuple then ω(T ) ⊂ σ Tb (T ).
Proof. Suppose without loss of generality that 0 /∈ σ Tb (T ). Then T is Taylor invertible
and 0 ∈ isoσ T (T ). So there exists a projection P ∈ B(H) satisfying that
(i) P commutes with T i (i = 1, · · · , n);
(ii) σ T (T | P (H) ) = {0} and σ T (T | (I−P )(H) ) = σ T (T ) \ {0};
(iii) P is of finite rank
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