#topologies on the set of bounded operators on some Hilbert space

[mark-gently]

tfw the ultraweak topology is stronger than the weak topology

[kingviquel]

So I’ve figured out some of the details revolving around the question of isometry groups, let’s try to put this into a nice summary :

- The compact-open topology makes composition and evaluation of continuous maps (between suitable spaces) continuous.

- However tempting though, the group of homeomorphisms doesn’t become a topological group for that topology, even for very regular spaces (I’m thinking locally compact metric spaces). Sure, composition of maps will be continuous, and the natural action will be continuous, but inversion is not continous in general. This is a good exercise : convince yourself that the inversion map Homeo(R) --> Homeo(R) is not continous for the compact-open topology. In some sense, the issue is that there are too many homeomorphisms in general.

- The definitive result is the following : the isometry group of a locally compact metric space is a (Hausdorff, locally compact) topological group and the natural action is continuous. Inversion becomes a continuous map, and it is best seen using the topology induced by uniform convergence on compact subsets (which is equivalent to the compact-open topology in this setting).

- Applying the previous result yields a very large collection of interesting topological groups, namely isometry groups of discrete spaces (those are the discrete symmetric groups), isometry groups of locally finite graphs, isometry groups of riemannian manifolds (those are Lie groups, but this is much harder to show). And most groups that I know can be embedded in (at least) one of these collections, so it really makes sense to focus on them.

- Now infinite dimensional Hilbert spaces are a typical example of good metric spaces which are not locally compact, so the previous theorem fails to be applicable. The crucial point is that the compact-open topology doesn’t necessarily make composition of maps continuous anymore.

- Now we are (or at least, I am) interested in unitary groups U(H). There are three natural topologies for spaces of operators : the norm topology (explicit), the strong topology (uniform convergence on compact subsets) and the weak topology (pointwise convergence). When H is finite dimensional, these are all the same (because the unit ball is compact). As soon as H is infinite dimensional, they need not be the same. The natural inclusions are norm ==> strong ==> weak.

- The definitive result is the following : firstly, with the norm topology, U(H) is a topological group. Secondly, the strong and weak topology are the same (unitary operators have norm 1, so any family is uniformly bounded), and it makes U(H) into a locally compact topological group. Thirdly, when H is infinite dimensional, U(H) is not locally compact for the norm topology.

- One remark : when reading about unitary representations, you’ll find that it is standard to restrict to so-called “strongly continuous” representations. This amounts to saying that the representation is a continuous homomorphism, where U(H) is equipped with the strong topology. This allows the associated group action to be continuous in the strong sense : for every vector in H, the action is continuous.

- One last remark : the Mazur-Ulam theorem says that isometries of real Hilbert spaces are linear. This allows to prove that the total isometry group of an affine Hilbert space is simply the semi-direct product H x U(H).

[thesupergamercorpus]

Today I wanted to write about one of my ‘heroes’ or inspirations. The way I look at them is somewhat like this quote:

“Trophies and great men are not only to be gazed upon, but also inspire to do the same.”

I will write down one of the best stories I have read about John von Neumann. But first of all, this is the list he is known for. I will make the ones I think are most popular bold:

Abelian von Neumann algebra, Affiliated operator, Amenable group, Arithmetic logic unit, Artificial viscosity, Axiom of regularity, Axiom of limitation of size, Backward induction, Blast wave (fluid dynamics), Bounded set (topological vector space), Carry-save adder, Cellular automata, Class (set theory), Computer virus, Commutation theorem, Continuous geometry, Coupling constants, Decoherence theory (quantum mechanics), Density matrix, Direct integral, Doubly stochastic matrix, Duality Theorem, Durbin–Watson statistic, EDVAC, Ergodic theory, Explosive lenses, Game theory, Hilbert’s fifth problem, Hyperfinite type II factor, Inner model, Inner model theory, Interior point method, Koopman–von Neumann classical mechanics, Lattice theory, Lifting theory, Merge sort, Middle-square method, Minimax theorem, Monte, Carlo method, Mutual assured destruction, Normal-form game, Operation Greenhouse, Operator theory, Pointless topology, Polarization identity, Pseudorandomness, Pseudorandom number generator, Quantum logic, Quantum mutual information, Quantum statistical mechanics, Radiation implosion, Rank ring, Self-replication, Software whitening, Sorted array, Spectral theory, Standard probability space, Stochastic computing, Stone–von Neumann theorem, Subfactor, Ultrastrong topology, Von Neumann algebra, Von Neumann architecture, Von Neumann bicommutant theorem, Von Neumann cardinal assignment, Von Neumann cellular automaton, Von Neumann interpretation, Von Neumann measurement scheme, Von Neumann ordinals, Von Neumann universal constructor, Von Neumann entropy, Von Neumann Equation, Von Neumann neighborhood, Von Neumann paradox, Von Neumann regular ring, Von Neumann–Bernays–Gödel set theory, Von Neumann universe, Von Neumann spectral theorem, Von Neumann conjecture, Von Neumann ordinal, Von Neumann’s inequality, Von Neumann’s trace inequality, Von Neumann stability analysis, Von Neumann extractor, Von Neumann ergodic theorem, Von Neumann–Morgenstern utility theorem, ZND detonation model

Cellular automata → this one appears to function and replicate like DNA. Cellular automata preceded the discovery of the structure of DNA.

Decoherence theory (quantum mechanics) → quantum states get continuously ‘pushed around’ by external influences (like being observed e.g. the double-slit experiment), which can change their original state. A quantum state resides in a ‘superposition’. Superposition simply means a state where two or more ‘states’ are combined, like an up and down state simultaneously. When that is the case, a quantum system resides in coherence. When observing that system, decoherence, or wave function collapse happens e.g. the original quantum system both had an up and down state simultaneously, but after being observed, now only has either an up state or a down state.

Merge sort → see the chapter 08/31/2019—Top-down, bottom-up thinking, sorting algorithms, and working memory where I discuss this computer sorting algorithm and combine it with top-down and bottom-up thinking.

Self-replication → a machine replicating itself. If machines are also able to upgrade themselves with each replication, a so-called technological singularity can occur (Google it).

Von Neumann architecture → essentially how our computers are built.

Now onto some stories of him. Most information is taken from Wikipedia.

Examination and Ph.D.

He graduated as a chemical engineer from ETH Zurich in 1926 (although Wigner says that von Neumann was never very attached to the subject of chemistry), and passed his final examinations for his Ph.D. in mathematics simultaneously with his chemical engineering degree, of which Wigner wrote, “Evidently a Ph.D. thesis and examination did not constitute an appreciable effort.”

Mastery of mathematics

Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: “Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods.” He went on to explain that the three methods were:

A facility with the symbolic manipulation of linear operators;

An intuitive feeling for the logical structure of any new mathematical theory;

An intuitive feeling for the combinatorial superstructure of new theories.

Edward Teller wrote that “Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique.”

Cognitive abilities

As a six-year-old, he could divide two eight-digit numbers in his head and converse in Ancient Greek. When he was sent at the age of 15 to study advanced calculus under analyst Gábor Szegő, Szegő was so astounded with the boy’s talent in mathematics that he was brought to tears on their first meeting.

Hans Bethe on von Neumann

Nobel Laureate Hans Bethe said “I have sometimes wondered whether a brain like von Neumann’s does not indicate a species superior to that of man”, and later Bethe wrote that “von Neumann’s brain indicated a new species, an evolution beyond man”.

Edward Teller

Edward Teller admitted that he “never could keep up with John von Neumann.”

Teller also said “von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us.”

George Dantzig

George Dantzig is the mathematician who thought that two problems on the blackboard were homework. He solved them and handed them, albeit a bit later, so he thought they were overdue.

Here’s the plot twist: They were two famous unsolved problems in statistics with which the mathematics community struggled for decades.

When George Dantzig brought von Neumann an unsolved problem in linear programming “as I would to an ordinary mortal”, on which there had been no published literature, he was astonished when von Neumann said “Oh, that!” before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceived theory of duality.

Johnny as a student

George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said “Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he’d come to me at the end of the lecture with the complete solution scribbled on a slip of paper.”

Nobel Prizes

Peter Lax wrote, “To gain a measure of von Neumann’s achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a ​3 1⁄2-fold winner, for his work in physics, in particular, quantum mechanics”.

von Neumann as a teacher

Von Neumann was the subject of many dotty professor stories. He supposedly had the habit of simply writing answers to homework assignments on the board (the method of solution being, of course, obvious). One time one of his students tried to get more helpful information by asking if there was another way to solve the problem. Von Neumann looked blank for a moment, thought, and then answered, “Yes.”

Henry Ford

Henry Ford had ordered a dynamo for one of his plants. The dynamo didn’t work, and not even the manufacturers could figure out why. A Ford employee told his boss that von Neumann was “the smartest man in America,” so Ford called von Neumann and asked him to come out and take a look at the dynamo.

Von Neumann came, looked at the schematics, walked around the dynamo, then took out a pencil. He marked a line on the outside casing and said, “If you’ll go in and cut the coil here, the dynamo will work fine.”

They cut the coil, and the dynamo did work fine. Ford then told von Neumann to send him a bill for the work. Von Neumann sent Ford a bill for $5,000. Ford was astounded – $5,000 was a lot in the 1950s – and asked von Neumann for an itemised account. Here’s what he submitted:

Drawing a line with the pencil: $1

Knowing where to draw the line with the pencil: $4,999

Ford paid the bill.

David Blackwell

Blackwell did a year of postdoctoral research as a fellow at the Institute for Advanced Study in 1941 after receiving a Rosenwald Fellowship. There he met John von Neumann, who asked Blackwell to discuss his Ph.D. thesis with him. Blackwell, who believed that von Neumann was just being polite and not genuinely interested in his work, did not approach him until von Neumann himself asked him again a few months later. According to Blackwell, “He (von Neumann) listened to me talk about this rather obscure subject and in ten minutes he knew more about it than I did.”

von Neumann was the only genius

Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Lutheran Church, the majority of its pupils were Jewish. The school system produced a generation noted for intellectual achievement. Wigner was a year ahead of von Neumann at the Lutheran School. When asked why the Hungary of his generation had produced so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that von Neumann was the only genius.”

[eatsleepmath]

Hilbert spaces, 2

We continue with Hilbert space theory. This was fairly fun to learn about.

Proposition. (Parseval’s identity) A subset \(\{e_j \operatorname{\big{|}} j \in J\}\) of a Hilbert space \(\mathfrak{H}\) is said to be orthonormal if \(||e_j|| = 1\) for every \(j\), and \((e_j, e_i) = 0\) for all \(i \neq j\). Furthermore, if the subspace spanned by the family \(\{e_j \operatorname{\big{|}} j \in J\}\) is dense in \(\mathfrak{H}\), we call it an orthonormal basis. This implies \(\mathfrak{H}\) is the orthogonal sum of the one-dimensional subspaces \(\mathbb{F} e_j\). Consequently, each element \(x \in \mathfrak{H}\) has the form \[x = \sum \alpha_j e_j\] where the sum converges in the \((2)\)-norm. Taking inner products by the \(e_j\)’s, we see that the coordinates for \(x\) are determined by \(\alpha_j \overset{\operatorname{df}}{=} (x | e_j).\) Computing \((x|x) = ||x||^2\) yields \[||x||^2 = \sum | \alpha_j |^2,\] which is Parseval’s identity.

Proposition. Every orthonormal set in a Hilbert space \(\mathfrak{H}\) can be enlarged to an orthonormal basis for \(\mathfrak{H}\).

Proof. Use Zorn’s lemma to obtain a maximal orthonormal set. Why is this a basis? If it didn’t span the entire set, there is a unit vector in its orthogonal complement. \(\square\)

Proposition. For every \(T \in \mathbf{B}(\mathfrak{H})\) we have \[\ker T^* = \left( T(\mathfrak{H}) \right)^{\perp}.\]

Proof. From the defining identity \((Tx | y) = (x | T^*y)\) we see that if \(y \in \ker T^*\), then \(y \in (T (\mathfrak{H}))^{\perp}\). Conversely, if \(y \in \left(T (\mathfrak{H}) \right)^{\perp}\), then \(T^* y \in \mathfrak{H}^{\perp} = 0.\) \(\square\)

Now we delve into compact operators.

Definition. An operator \(T\) on an infinite-dimensional Hilbert space \(\mathfrak{H}\) as finite rank if its image is a finite-dimensional subspace of \(\mathfrak{H}\) (hence closed.)

Here are some remarks on this definition:

The set of finite rank operators in \(\mathbf{B}(\mathfrak{H})\) is a subspace, in fact a subalgebra (in fact an ideal) of \(\mathbf{B}(\mathfrak{H})\).

If \(T\) is finite rank, we obtain an orthogonal decomposition \(\mathfrak{H} = T(\mathfrak{H}) \oplus \ker T^*\) which shows that \(T^*(\mathfrak{H}) = T^* T(\mathfrak{H})\), so that \(T^*\) has finite rank. Thus, \(\mathbf{B}_f(\mathfrak{H})\) is a self-adjoint ideal in \(\mathbf{B}(\mathfrak{H})\).

We will see that the class \(\mathbf{B}_f(\mathfrak H)\) will play a similar role as the space of compactly-supported continuous functions on a locally compact Hausdorff space when considering the bounded continuous functions out of a Hausdorff space; these classes will describe local phenomena on \(\mathfrak{H}\) and \(X\).

Passing to a limit in norm may destroy the exact “locality”, but enough structure is preserved to make these “quasilocal” operators and functions useful; we will study the closure of \(\mathbf{B}_f(\mathfrak H)\) in this section as a noncommutative analogue of \(C_0(X)\) in function theory.

Lemma. There is a net \((P_{\lambda})_{\lambda \in \Lambda}\) of projections in \(\mathbf{B}_f(\mathfrak{H})\) such that \(\) || P_ x - x || 0 \(\)

Proof. Take an orthonormal basis \(\{e_j \operatorname{\big{|}} j \in J\}\) for \(\mathfrak{H}\) and let \(\Lambda\) be the finite subsets of \(J\) ordered under inclusion.

For each \(\lambda \in \Lambda\), let \(P_{\lambda}\) be the projection of \(\mathfrak{H}\) onto the subspace \(\operatorname{span} \{e_j \operatorname{\big{|}} j \in \lambda\}\), so that \((P_{\lambda})_{\lambda \in \Lambda}\) is a net in \(\mathbf{B}_f(\mathfrak{H})\). If \(x \in \mathfrak{H}\), we have \(x = \sum \alpha_j e_j\), whence \[||P_{\lambda} x - x ||^2 = \sum |\alpha_j|^2\] the summation being over all \(j \not \in \lambda\) and this this tends to zero by the Parseval identity. \(\square\)

Definition. Recall that the weak topology on a Hilbert space \(\mathfrak{H}\) is the initial topology induced by the family of functionals \[\left\{(- | y)\right\}_{y \in \mathfrak{H}},\] hence is the weak-star topology on \(\mathfrak{H}\) pulled back to \(\mathfrak{H}\) by the conjugate linear isometry \(\Phi\).

Theorem. Let \(B\) be the closed unit ball in a Hilbert space \(\mathfrak{H}\). Then the following conditions on an operator \(T \in \mathbf{B}(\mathfrak{H})\) are equivalent:

\(T \in \left(\mathbf{B}_f(\mathfrak{H})\right)^{=}\)

\(T \operatorname{\restriction} B\) is a weak-norm continuous function \(B \to \mathfrak{H}\).

\(T(B)\) is compact in \(\mathfrak{H}\).

\((T(B))^{=}\) is compact in \(\mathfrak{H}\).

Each net in \(B\) has a subset whose image under \(T\) converges in \(\mathfrak{H}\).

Proof.

(i) \(\implies\) (ii):

Let \((x_{\lambda})_{\lambda \in \Lambda}\) be a weak-convergent net in \(B\) with limit \(x\). For each \(\epsilon > 0\), there is by our assuming \(\textbf{(i)}\) an \(S\) in \(\mathbf{B}_f(\mathfrak{H})\) with \(||S - T || < \epsilon / 3\), so that \[||T x_{\lambda} - T x || \leq 2 || T - S || + || S x_{\lambda} - S x || \leq \frac{2}{3} \epsilon + || S x_{\lambda} - S x ||.\] Since every operator in \(\mathbf{B}(\mathfrak{H})\) is weak-weak continuous, \(S x_{\lambda}\) weak-converges to \(Sx\). Since all vector space topologies coincide on finite-dimensional subspaces, \(S x_{\lambda}\) norm-converges to \(Sx\) as well.

Hence (eventually(?)) \(||T x_{\lambda} - T x || < \epsilon\), and since epsilon was arbitrary we have that \(T\) is weak-norm continuous.

(ii) \(\implies\) (iii):

The unit ball \(B\) is weakly compact; since \(T\) is continuous from \((\mathfrak{H}, w)\) to \((\mathfrak{H}, ||\cdot||)\) and the image of compact under continuous is compact, \(T(B)\) is norm-compact.

(iii) \(\implies\) (iv):

\(T(B)\) is closed whence norm-compactness and the fact that the norm topology is Hausdorff, whence compact subsets of Hausdorff spaces are closed.

(iv) \(\implies\) (v)

Since compactness is equivalent to all subnets having a convergent subnet, this is clear.

(v) \(\implies\) (i):

Take \((P_{\lambda})_{\lambda \in \Lambda}\) as in the previous lemma.

Then \(P_{\lambda} T \in \mathbf{B}_f(\mathfrak{H})\) for every \(\lambda \in \Lambda\), and we claim that \(P_{\lambda} T\) norm-converges to \(T\).

Suppose not. Then there is an \(\epsilon > 0\) (and refining \(\Lambda\) if necessary) there exists for every \(\lambda\) a unit vector \(x_{\lambda}\) such that \[||(P_{\lambda} T - T) x_{\lambda} || \geq \epsilon.\]

By assumption, we may assume that the net \((T x_{\lambda})_{\lambda \in \Lambda}\) is norm-convergent in \(\mathfrak{H}\) with limit \(y\).

By the lemma, \[\epsilon \leq ||(I - P_{\lambda})T x_{\lambda} || \leq ||(I - P_{\lambda})(T x_{\lambda} - y)|| + ||(I - P_{\lambda}) y ||\] \[\leq || T x_{\lambda} - y|| + ||(I - P_{\lambda}) y || \to 0,\] a contradiction since \(P_{\lambda}\) approximated the identity; thus \(||P_{\lambda} T - T|| \to 0\), as required.

\(\square\)

Definition. The class of operators satisfying any of the conditions above are called the compact operators, denoted \(\mathbf{B}_0(\mathfrak{H})\) to signify that they “vanish at infinity” (this notation is not standard, and occasionally the notation \(\mathbf{K}(\mathfrak{H})\) or \(\mathbf{C}(\mathfrak{H})\) is used.)

Remark. From condition \(\textbf{(i)}\), we see that \(\mathbf{B}_0(\mathfrak{H})\) is a norm-closed, self-adjoint ideal in \(\mathbf{B}(\mathfrak{H})\) (and actually the smallest such; for separable Hilbert spaces, the only closed ideal.)

Note that when \(\mathfrak{H}\) is infinite-dimensional, the identity operator is not compact. However, (c.f. the proof of \(\textbf{(v)} \implies \textbf{(i)}\)) \(\mathbf{B}_0(\mathfrak{H})\) has an approximate unit consisting of projections of finite rank.

Lemma. A diagonalizable operator \(T\) in \(\mathbf{B}(\mathfrak{H})\) is compact if and only if its eigenvalues \(\{\lambda_j \operatorname{\big{|}} j \in J\}\) corresponding to an orthonormal basis \(\{e_j \operatorname{\big{|}} j \in J\}\) belongs to \(c_0(J)\).

Proof. We can write \[Tx = \sum \lambda_j(x | e_j) e_j\] for every \(x \in \mathfrak{H}\). If \(T \in \mathbf{B}_0(\mathfrak{H})\) and \(\epsilon > 0\) are given, let \[J_{\epsilon} \overset{\operatorname{df}}{=} \{j \in J \operatorname{\big{|}} |\lambda_j| \geq \epsilon\}.\] If \(J_{\epsilon}\) is infinite, the net \((e_j)_{j \in J_{\epsilon}}\) will converge weakly to zero for any well-ordering of \(J_{\epsilon}\) because by Parseval’s identity, \((e_J | x) \to 0\) by Parseval’s identity.

Since \(||T e_j|| = |\lambda_j| \geq \epsilon\) for \(j \in J_{\epsilon}\), this contradicts condition \(\textbf{(ii)}\) (weak-norm continuity of \(T\) restricted to \(B\)) from the characterization of compact operators above.

Hence, \(J_{\epsilon}\) is finite for each \(\epsilon > 0\), so that the \(\lambda_j\)’s vanish at infinity.

Conversely, if \(J_{\epsilon}\) is finite for each \(\epsilon > 0\), we let \[T_{\epsilon} = \sum_{j \in J_{\epsilon}} \lambda_J ( \cdot \operatorname{\big{|}} e_j)e_j;\] \(T\) has finite rank and furthermore, \[||(T - T_{\epsilon})x||^2 = || \sum_{j \in J_{\epsilon}} \lambda_J (e | e_j) e_j ||^2\]

\[= \sum_{j \not \in J_{\epsilon}} |\lambda_j|^2|(x | e_j)|^2 \leq \epsilon^2||x||^2.\]

Thus, \(||T - T_{\epsilon}|| \leq \epsilon\), so \(T \in B_0(\mathfrak{H})\) by condition \(\textbf{(i)}\) of the characterization theorem. \(\square\)

Recall that a normal operator \(T\) is one which commutes with its adjoint.

Lemma. If \(x\) is an eigenvector for a normal operator \(T\), corresponding to the eigenvalue \(\lambda\), then \(x\) is an eigenvector for \(T^*\), corresponding to the eigenvalue \(\overline{\lambda}\). Eigenvectors for \(T\) corresponding to different eigenvalues are orthogonal.

Lemma. Every normal compact operator \(T\) on a complex Hilbert space \(\mathfrak{H}\) has an eigenvalue \(\lambda\) with \(|\lambda| = ||T||\).

Proof. Let \(B\) be the unit ball of \(\mathfrak{H}\). By \((3.3.3)\), \(T : B \to B\) is weak-norm continuous.

Hence, if \((x_i)\) weak-converges to \(x\) in \(B\), \[|(T x_i | x_i) - (Tx | x)| = |(T(x_i - x)| x_i) + (Tx | x_i - x)|\] \[\leq ||T(x_i - x)|| + |(T x | x_i - x)| \to 0,\] hence the function \[x \mapsto |T(Tx | x)|\] i weak-continuous on \(B\); since \(B\) was weakly compact, this function attains its maximum, and it can be seen (3.2.25) that this maximum must be precisely \[||T||.\] Hence for some \(x \in B\), \(|(Tx | x)| ||T||\). Since \[||T|| = |(Tx | x)| \leq ||Tx||||x|| \leq ||T||,\] \(|(Tx | x)| = ||Tx || || x||.\) Since here the Cauchy-Schwarz inequality has become an equality, \(Tx\) and \(x\) are proportional, so that \(Tx = \lambda x\) for some \(\lambda\), with \(|\lambda| = ||T||\). \(\square\)

Theorem. Every normal compact operator \(T\) on a complex Hilbert space \(\mathfrak{H}\) is diagonalizable and its eigenvalues counted with multiplicity vanish at infinity; conversely, each such operator is normal and compact.

Proof. It suffices to show that every normal compact operator is diagonlizable, since then we can invoke the lemma \((3.3.5)\) to get the rest of the theorem.

Apply Zorn’s lemma to the poset of orthonormal systems in \(\mathfrak{H}\) consisting of eigenvectors for \(T\) to obtain a maximal orthonormal system of eigenvectors \(\{e_j | j \in J\}\), with corresponding family \(\{\lambda_j | j \in J\}\) the corresponding family of eigenvalues.

Let \(P\) be the projection onto the span of the \(e_j\)’s. We want to show that \(P\) is in fact the identity, i.e. that the \(e_j\)’s are an orthonormal basis for all of \(\mathfrak{H}\).

For each \(x \in \mathfrak{H}\), we have (by 3.3.6): \[T Px = T \sum (x | e_j) e_j = \sum (x | e_j)\lambda_j e_j\]

\[= \sum (x | \overline{\lambda}_j e_j)e_j = \sum(x | T^* e_j) e_j\]

\[= \sum (Tx | e_j) e_j = P Tx.\]

Since we see that \(T\) and \(P\) commute, the operator \((I - P)T\) is normal and compact. If \(P \neq I\), either \((I - P)T = 0\) or \((I - P)T \neq 0\).

In the first case, every unit vector \(e_0\) in \((I - P)(\mathfrak{H})\) is an eigenvector for \(T\). In the second case by (3.3.7) there is a unit vector \(e_0\) in \((I - P) \mathfrak{H}\) with \(T e_0 = \lambda e_0\) and \(|\lambda| = ||(I - P)T||.\)

In either case this contradicts the maximality of \(\{e_j | j \in J\}\); therefore, \(P = I\). \(\square\)

Definition. We define \(x \odot y\) to be the rank-one operator in \(\mathbf{B}(\mathfrak{H})\) determined by the vectors \(x\) and \(y\) in \(\mathfrak{H}\) by the formula \[(x \odot y)z \overset{\operatorname{df}}{=} (z | y)x.\] Note that \((- \odot - ) : \mathfrak{H} \times \mathfrak{H} \to \mathbf{B}_f(\mathfrak{H})\) is a sesquilinear map.

If \(||e|| = 1\) is a unit vector, then \(e \odot e\) is the one-dimensional projection of \(\mathfrak{H}\) onto \(\mathbb{C} e\). Every normal compact operator on \(\mathfrak{H}\) can now (by 3.3.8) be written in the form \[T = \sum \lambda_j e_j \odot e_j\] for a suitable orthonormal basis \((e_j)_{j \in J}\).

Why does the sum converge in norm? Well, either the terms in the series are only finitely supported; if it’s not finitely supported, it’s countably infinite, and then the sequence \(\lambda_j \to 0\).

Definition. We say that the compact set \(\operatorname{sp}(T) \overset{\operatorname{df}}{=} \{\lambda_j \operatorname{\big{|}} j \in J_0\} \cup \{0\}\) is the spectrum of \(T\).

For every continuous function \(f\) on \(\operatorname{sp}(T)\) we define: \[f(T) = \sum f(\lambda_j) e_j \odot e_j.\] Then \(f(T)\) is compact if and only if \(f(0) = 0\), and the map \(f \to f(T)\) is an isometric \(*\)-preserving homomorphism of \(C(\operatorname{sp}(T)\) into \(\mathbf{B}(\mathfrak{H})\). Moreover, if \(f(z) = \sum \alpha_{n,m} z^n \overline{z}^m,\) a polynomial in the two commuting variables \(z\) and \(\overline{z}\), then \[f(T) = \sum \alpha_{n,m} T^n T^{*m}.\]

This is the spectral mapping theorem for normal, compact operators; in section \(3.4\) we will see a generalized version of this theorem, valid for every normal operator.

Definition. Since \(\mathbf{B}_0(\mathfrak{H})\) is a closed ideal in \(\mathbf{B}(\mathfrak{H})\), the quotient \(\mathbf{B}(\mathfrak{H})/\mathbf{B}_0(\mathfrak{H})\) is a Banach algebra1 under the quotient norm \((4.1.2)\). (Actually, by \(4.3.14\) the quotient is even a C-star algebra.) This quotient is called the Calkin algebra, and several properties of \(\mathbf{B}(\mathfrak{H})\) can be conveniently formulated in terms of the Calkin algebra.

If \(S\) and \(T\) are elements in \(\mathbf{B}(\mathfrak{H})\) and \(S \sim T\) modulo a compact operator, we say that \(S\) is a compact perturbation of \(T\), and this just means that \(S\) and \(T\) are identified after being sent to the Calkin algebra.

Such “local” perturbations occur frequently in applications, and properties of an operator that are invariant under compact perturbations are therefore highly valuable. Such an example is the index.

Recall that Banach algebras are \(\mathbb{F}\)-algebras whose underlying \(\mathbb{F}\)-vector space is Banach, such that multiplication is norm-continuous (this is equivalent to requiring the norm to be submultiplicative.)↩