Since any norm onC(X) can be associated with a semi-norm on B(X) with kernel
K(X), and vice versa (refer to§1.2), we seek a definition of a semi-norm onB(X)
26 Chapter 2. An Alternative Algebra Norm on C(X) whose value for an operatorT will be 0 if and only ifT is compact. Such a semi-norm should thus involve a measure of how close an operator is to being compact, but we must use a quantity that is sufficiently different to the essential norm in order that the new definition might not be equivalent. To achieve this, we consider the formulation of compactness in terms of the presence of convergent subsequences amongst the im- ages of bounded sequences. In the following, Definition 2.1.1, Proposition 2.1.3, and its proof, were inspired by [Fer, Defn. 2 and Lem. 2], which drew similar conclusions about an analogous norm on quotient algebras of the formB(X)/S(X).
2.1.1 Definition. LetXandY be Banach spaces. For (xn)⊂XandT ∈B(X, Y), set dT ,(xn)e = lim N→∞ sup n1,n2≥N kT(xn1 −xn2)k.
Forλ >0, denote by ∆λX the collection of all sequences in X of diameter at mostλ, that is, ∆λX = {(xn)⊂X: (∀n1, n2∈N) kxn1−xn2k ≤λ}. Now define kTkK = sup (xn)⊂∆1X inf (xnk)⊂(xn) dT ,(xnk)e.
2.1.2 Remark. Note that kTkK ≤ kTk for all T ∈B(X, Y). The quantity
dT ,(xn)e is in some sense a measure of the divergence of the sequence (T xn), and equals 0 if and only if (T xn) converges. As such, the following facts regardingd·,·e are not surprising. GivenT ∈B(X, Y) and (xn)⊂X, suppose (xnk)⊂(xn). Then,
comparingdT ,(xnk)etodT ,(xn)e, for a givenN ∈Nthe supremum in the definition
ford·,·eranges over a restricted subset, so it follows that
dT ,(xnk)e ≤ dT ,(xn)e .
Also, note that removing a finite number of members from (xn) will not change the value of the limit in the definition of dT ,(xn)e, so,
dT ,(xn)∞n=Me = dT ,(xn)e
for all M ∈N.
2.1 Another way to measure the ‘compactness’ of an operator 27
in mind, it follows that the quantity inf
(xnk)⊂(xn)
dT ,(xnk)e
measures how ‘close’ (T xn) is to having a convergent subsequence. Thus, the outer supremum in the definition (2.1.1) ofkTkK detects the maximum ‘distance’ images of sequences inT(∆1X) are from having convergent subsequences.1 This provides the intended conceptual basis for the following elementary properties ofk · kK.
2.1.3 Proposition. Let X and Y be Banach spaces. Then the function k · kK is a
semi-norm on B(X, Y). Furthermore, its kernel is the space of compact operators
K(X, Y).
Proof. We will identify the kernel of k · kK first. Since an operator T ∈B(X, Y) is compact if and only if every bounded sequence (xn) in X has a subsequence (xnk)
such that the sequence (T xnk)⊂Y converges, it is immediate from Definition 2.1.1
that ifT is compact thenkTkK = 0. Conversely, ifkTkK = 0 then we have that
inf
(xnk)⊂(xn)
dT ,(xnk)e = 0
for all (xn) ∈ ∆1X. A diagonalisation argument then shows that this infimum is actually attained, so that each (xn) ∈ ∆1X has a subsequence (xnk) such that the
sequence (T xnk)⊂Y converges (the argument will not be given here since, in a more
general context, it forms part of the proof of Lemma 2.2.2 below). HencekTkK = 0 only if T is compact, and so the kernel ofk · kK is K(X, Y).
To show thatk · kK is a semi-norm, it suffices to check the triangle inequality; clearlykcTkK =|c|kTkK forc∈Fand T ∈B(X, Y). Let (xn)⊂X be an arbitrary sequence and note that d·,(xn)e satisfies the triangle inequality (apply the triangle inequality for the normk · kof Y to the respective part of Definition 2.1.1). Now let T andU be elements ofB(X, Y) and suppose (xn)∈∆1X is such that the supremum in the definition ofkT+UkK is attained up to a givenε >0. It follows that
kT +UkK ≤ dT +U ,(xnk)e+ε ≤ dT ,(xnk)e+dU ,(xnk)e+ε (2.1.3a)
for all (xnk)⊂(xn).
1
Using ∆1X as the canonical source of bounded sequences inX is more natural than usingBX, in this setting.
28 Chapter 2. An Alternative Algebra Norm on C(X) Now let (xn1 k)⊂(xn) be such that inf (xnk)⊂(xn) dT ,(xnk)e
is attained up toε. By Remark 2.1.2, for all (xn2
k)⊂(xn1k) we have l T ,(xn2 k) m ≤ lT ,(xn1 k) m ,
so the above infimum will also be attained up to ε by every (xn2
k) ⊂ (xn
1
k). Thus
from (2.1.3a) it follows that
kT +UkK ≤ lT ,(xn2 k) m +lU ,(xn2 k) m +ε ≤ inf (xnk)⊂(xn) dT ,(xnk)e+ε+ l U ,(xn2 k) m +ε ≤ kTkK+ l U ,(xn2 k) m + 2ε for all (xn2 k)⊂(xn 1 k). Hence kT +UkK ≤ kTkK+ inf (xn2 k )⊂(xn1 k) l U ,(xn2 k) m + 2ε ≤ kTkK+kUkK+ 2ε .
This holds for allε >0, therefore
kT +UkK ≤ kTkK+kUkK.
So, if we restrict attention to operators inB(X) for some Banach spaceX and naturally identify k · kK with a norm on C(X), we have that k · kK is a potential candidate for a new algebra norm. It remains to check thatk · kK is submultiplica- tive. This is the conclusion of the next section, which also amasses some technical facts of later importance.