It is elementary that finite spectrum normal operators are diagonalizable. However, one may ask about the possibility of diagonalization relative to a fixed orthonormal basis (or atomic masa) by a unitary of the form \(U = I+K\) where \(K\) lies in a given proper operator ideal \(\mathcal{J}\text{.}\) For this we use the term restricted diagonalization. This concept has been studied by others in the aforementioned paper of Brown–Douglas–Fillmore [5], as well as by Beltiţă–Patnaik–Weiss [8], and Hinkkanen [11]. To our knowledge, the term restricted diagonalization was introduced by Beltiţă–Patnaik–Weiss.
The next result is a corollary of Proposition 2.3, Proposition 2.4. It describes the conditions under which a projection experiences restricted diagonalization. In the special case of the Hilbert–Schmidt ideal, this corollary shows that it suffices to examine the diagonal of the projection.
For a projection \(P\) and a proper operator ideal \(\mathcal{J}\text{,}\) the following are equivalent:
If \(\mathcal{J} = \mathcal{C}_2\text{,}\) then these are also equivalent to:
i \(\Rightarrow\) ii. Suppose that \(P\) is diagonalizable by a unitary \(U=I+K\) with \(K \in \mathcal{J}\text{.}\) Then setting \(Q := UPU^{*}\text{,}\) we have \(P-Q = -KP-PK^{*}-KPK^{*} \in \mathcal{J}\text{.}\)
ii \(\Rightarrow\) i. Suppose \(Q\) is a diagonal projection for which \(P-Q \in \mathcal{J}\text{.}\) By replacing \(Q\) with a diagonal projection \(Q'\) that is a finite perturbation of \(Q\text{,}\) we can assume that \([P:Q] = 0\text{.}\) Indeed, notice that if \([P:Q] \lt 0\text{,}\) then \(\trace Q \ge -[P:Q]\text{,}\) so there is a diagonal subprojection \(Q'\) of \(Q\) with \(\trace(Q-Q') = -[P:Q]\text{.}\) Similarly, if \([P:Q] > 0\text{,}\) then \(\trace Q^{\perp} \ge [P:Q]\text{,}\) so there is a diagonal subprojection \(R\) of \(Q^{\perp}\) with \(\trace R = [P:Q]\text{,}\) and in this case we set \(Q' = Q+R\text{.}\) In either case, the construction guarantees \([P:Q] = -[Q:Q']\text{,}\) and hence by Proposition 2.2iii, \([P:Q'] = [P:Q] + [Q:Q'] = 0\text{.}\) Therefore by Proposition 2.3, \(P\) and \(Q'\) are conjugated by a unitary \(U=I+K\) with \(K \in \mathcal{J}\text{,}\) and hence \(P\) is diagonalized by \(U\text{.}\)
ii \(\Rightarrow\) i. If \(P-Q \in \mathcal{C}_2\text{,}\) then by Proposition 2.4, for \(a,b\) defined as in that proposition, \(a+b \lt \infty\text{.}\) Equivalently, \((d_n) \in \Lim(\{0,1\})\text{.}\)
i \(\Rightarrow\) ii. If the diagonal \((d_n)\) of \(P\) lies in \(\Lim(\{0,1\})\text{,}\) then there are some choices \(x_n \in \{0,1\}\) for which \((d_n - x_n) \in \ell^1\text{.}\) Let \(Q\) be the diagonal projection onto the \(\spans\{ e_n \mid x_n =1 \}\text{.}\) Then for \(a,b\) as defined in Proposition 2.4, \(a+b \lt \infty\text{,}\) and so \(P-Q \in \mathcal{C}_2\) by that result.
We will generalize Corollary 3.1 to finite spectrum normal operators. The equivalence i \(\Leftrightarrow\) ii is generalized by Theorem 3.4, and i \(\Leftrightarrow\) i by Theorem 4.2.
Proposition 2.3 can be bootstrapped by induction to characterize when a pair of finite collections of mutually orthogonal projections can be simultaneously conjugated by a unitary \(U = I + K\) with \(K \in \mathcal{J}\text{.}\)
Suppose \(\{P_k\}_{k=1}^m, \{Q_k\}_{k=1}^m\) are each finite sets of mutually orthogonal projections, and \(\mathcal{J}\) is a proper operator ideal. Then there is some unitary \(U = I+K\) with \(K \in \mathcal{J}\) for which \(Q_k = UP_kU^{*}\) for \(1 \le k \le m\) if and only if \(P_k - Q_k \in \mathcal{J}\) and \([P_k:Q_k] = 0\) for all \(1 \le k \le m\text{.}\)
One direction is straightforward. Namely, if there exists a unitary \(U = I+K\) with \(K \in \mathcal{J}\) for which \(Q_k = UP_k U^{*}\) for all \(1 \le k \le m\text{,}\) then by Proposition 2.3 \(P_k - Q_k \in \mathcal{J}\) and \([P_k:Q_k] = 0\text{.}\)
For the other direction, we use induction on \(m\text{,}\) and the base case \(m=1\) follows from Proposition 2.3. Let \(m \in \mathbb{N}\) and suppose that if \(\{P_k\}_{k=1}^m, \{Q_k\}_{k=1}^m\) are each sets of mutually orthogonal projections and satisfy \(P_k - Q_k \in \mathcal{J}\) and \([P_k:Q_k] = 0\text{,}\) then there is a single unitary \(U=I+K\) with \(K \in \mathcal{J}\) which conjugates \(P_k\) into \(Q_k\text{,}\) i.e., \(Q_k = UP_kU^{*}\text{.}\)
Now suppose we have two sets of \(m+1\) mutually orthogonal projections satisfying these conditions. By Proposition 2.3 there is a unitary \(V = I+K\) with \(K \in \mathcal{J}\) for which \(Q_{m+1} = VP_{m+1}V^{*}\text{.}\) Moreover, for \(1 \le k \le m\text{,}\) \(P'_k := VP_k V^{*}\) satisfies \(P_k - P'_k \in \mathcal{J}\) and \([P_k:P'_k] = 0\text{.}\) Therefore \(P'_k - Q_k \in \mathcal{J}\) and \([P'_k:Q_k] = 0\) by Proposition 2.2iii. Applying the inductive hypothesis to the collections \(\{P'_k\}_{k=1}^m, \{Q_k\}_{k=1}^m\) on the Hilbert space \(Q_{m+1}^{\perp} \Hil\) yields a unitary \(W = Q_{m+1}^{\perp} + K'\) acting on \(Q_{m+1}^{\perp} \Hil\) with \(K' \in \mathcal{J}\text{,}\) and which conjugates \(P'_k\) into \(Q_k\) for \(1 \le k \le m\text{.}\) Extending this to the unitary \(Q_{m+1} \oplus W\) acting on \(\Hil\) and setting \(U = (Q_{m+1} \oplus W)V\text{,}\) we find that \(U\) is of the desired form and \(UP_k U^{*} = Q_k\) for \(1 \le k \le m+1\text{.}\)
The following lemma weakens the sufficient condition of Lemma 3.2 so long as we are allowed to perturb the diagonal projections.
Suppose that \(\{P_k\}_{k=1}^m, \{Q_k\}_{k=1}^m\) are each collections of mutually orthogonal projections for which \(P_k - Q_k \in \mathcal{J}\) and \(\sum_{k=1}^m [P_k:Q_k] = 0\text{.}\) Then for every atomic masa \(\mathcal{A}\) containing \(\{Q_k\}_{k=1}^m\text{,}\) there exist mutually orthogonal projections \(\{Q'_k\}_{k=1}^m \subseteq \mathcal{A}\) for which \(P_k - Q'_k \in \mathcal{J}\) and \([P_k:Q'_k] = 0\text{.}\)
Suppose \(\{Q_k\}_{k=1}^m\) lies in an atomic masa. Note that such a masa always exists since this is a finite collection of mutually orthogonal (hence commuting) projections. The argument is by induction on \(m\text{.}\) When \(m=1\text{,}\) the claim is trivial.
Now suppose \(m > 1\text{.}\) Either \([P_k : Q_k] = 0\) for all \(k\) already, or there are two indices \(1 \le i,j \le m\) with \([P_i : Q_i] \lt 0 \lt [P_j:Q_j]\text{.}\) Notice that \(\trace Q_i \ge -[P_i : Q_i]\text{.}\) Let \(Q\) be a diagonal subprojection of \(Q_i\) with \(\trace Q = \min \{ -[P_i:Q_i], [P_j:Q_j] \}\text{.}\) Then we replace \(Q_i\) with \(Q_i - Q\) and \(Q_j\) with \(Q_j + Q\text{.}\) By construction, either \(\big[P_i:(Q_i - Q)\big] = 0\) or \(\big[P_j:(Q_j+Q)\big] = 0\text{.}\) So now we have \(n-1\) pairs of projections for which the sum of the essential codimensions is zero. By induction we can actually force them all to be zero while maintaining the condition that the \(Q'_k\) projections are diagonal.
Suppose \(\mathcal{J}\) is a proper operator ideal. A finite spectrum normal operator is diagonalizable by a unitary \(U=I+K\) with \(K \in \mathcal{J}\) if and only if each spectral projection differs from a diagonal projection by an element of \(\mathcal{J}\text{.}\)
Let \(N = \sum_{k=1}^m \lambda_k P_k\) be a finite spectrum normal operator with spectral projections \(P_k\) associated to the eigenvalues \(\lambda_k\text{.}\) One direction is trivial, namely, if \(N\) is diagonalizable by a unitary \(U=I+K\) with \(K \in \mathcal{J}\text{,}\) then the projections \(Q_k := UP_kU^{*}\) are diagonal and \(P_k - Q_k \in \mathcal{J}\text{.}\)
For the other direction, suppose that for each \(P_k\) there is a diagonal projection \(Q_k\) for which \(P_k - Q_k \in \mathcal{J}\text{.}\) The operators \(Q_jQ_k\) are projections because \(Q_j,Q_k\) are commuting projections. Then since \(P_jP_k = \delta_{jk}P_j\text{,}\) for \(j\not=k\) we obtain
\begin{equation} \begin{split} Q_jQ_k &= \big(P_j+(Q_j-P_j)\big)\big(P_k+(Q_k-P_k)\big) \\ &= (Q_j-P_j)P_k + P_j(Q_k-P_k) + (Q_j-P_j)(Q_k-P_k) \in \mathcal{J}. \end{split}\label{eq-Q_j-times-Q_k-in-J}\tag{3.1} \end{equation}Therefore \(Q_jQ_k\) are finite projections when \(j \not= k\text{.}\)
Now let \(Q'_1 := Q_1\) and inductively define \(Q'_j = Q_j-Q_j(Q'_1+\cdots+Q'_{j-1})\) for \(1 \lt j \lt m\) and finally \(Q'_m = I-(Q'_1+\cdots+Q'_{m-1})\text{.}\) It is clear that for \(1 \le j \lt m\text{,}\) \(Q'_j\) is in the algebra generated by \(\{Q_1,\ldots,Q_j\}\) and is therefore diagonal. Moreover, for \(1 \le j \lt m\text{,}\) by (3.1) and induction \(Q'_j - Q_j\) is finite rank, and hence \(P_j - Q'_j \in \mathcal{J}\text{.}\) Thus, \(Q'_m\) is a \(\mathcal{J}\)-perturbation of \(I-(P_1+\cdots+P_{m-1}) = P_m\text{,}\) and hence \(P_m - Q'_m \in \mathcal{J}\) as well. By Proposition 2.2(ii),
\begin{equation*} \sum_{k=1}^m [P_k:Q'_k] = \left[ \sum_{k=1}^m P_k: \sum_{k=1}^m Q'_k \right] = [I:I] = 0. \end{equation*}So, by Lemma 3.3, we may assume by passing to a possibly different collection of diagonal \(Q'_k\) that, in fact, \([P_k:Q'_k] = 0\) for \(1 \le k \le m\text{.}\) Finally, by Lemma 3.2 there is a unitary \(U=I+K\) with \(K \in \mathcal{J}\) for which \(Q'_k = UP_k U^{*}\) for each \(1 \le k \le m\text{.}\) Therefore, \(UNU^{*} = \sum_{k=1}^m \lambda_k Q'_k\text{,}\) which is a diagonal operator.
This subsection is motivated by the following observation about the condition \((d_n) \in \Lim\big(\spec(N)\big)\) in Arveson's theorem.
Let \(N\) be a normal operator with finite spectrum and let \((d_n)\) be the diagonal of \(N\text{.}\) Then \((d_n) \in \Lim\big(\spec(N)\big)\) if and only if there exists a diagonal operator \(N' = \diag(x_n)\) such that \(\spec(N') \subseteq \spec(N)\text{,}\) and \(E(N-N')\) is trace-class, in which case
\begin{equation} \trace\big(E(N-N')\big) = \sum_{n=1}^{\infty} (d_n - x_n).\label{eq-renormalized-sum-trace}\tag{3.2} \end{equation}(\(\Rightarrow\)) Suppose \((d_n) \in \Lim\big(\spec(N)\big)\text{.}\) Then there is a sequence \((x_n)\) with \(x_n \in \spec(N)\) such that \((d_n - x_n)\) is absolutely summable, and we may take \(N' := \diag(x_n)\text{.}\) Therefore, since \((d_n - x_n)\) is absolutely summable,
\begin{equation*} \trace \abs{E(N-N')} = \sum_{n=1}^{\infty} \abs{d_n-x_n} \lt \infty, \end{equation*}and hence \(E(N-N')\) is trace-class.
(\(\Leftarrow\)) Suppose \(N'\) is a diagonal operator with \(\spec(N') \subseteq \spec(N)\) and \(E(N-N')\) trace-class, and let \((x_n)\) denote the diagonal of \(N'\text{.}\) Then \(x_n \in \spec(N') \subseteq \spec(N)\) and since \(E(N-N')\) is trace-class,
\begin{equation*} \sum_{n=1}^{\infty} \abs{d_n-x_n} = \trace \abs{E(N-N')} \lt \infty. \end{equation*}Therefore \((d_n-x_n)\) is absolutely summable and hence \(d_n \in \Lim\big(\spec(N)\big)\text{.}\)
Notice that whenever either of the equivalent conditions is satisfied, we have the equality
\begin{equation*} \trace\big(E(N-N')\big) = \sum_{n=1}^{\infty} (d_n - x_n). \qedhere \end{equation*}The remainder of the section is devoted to analyzing the expression \(E(N-N')\) when \(N'\) is a restricted diagonalization of a normal operator \(N\) (not necessarily with finite spectrum), i.e., when \(N' = UNU^{*}\) where \(U = I + K\) is unitary and \(K \in \mathcal{J}\text{.}\)
As in [10], the arithmetic mean closure \(\mathcal{J}^{-}\) of an operator ideal \(\mathcal{J}\) is the set of operators \(T\) whose singular values are weakly majorized by the singular values of an operator \(A \in \mathcal{J}\text{;}\) that is, if \(s(T)\) denotes the singular value sequence of a compact operator \(T\text{,}\)
\begin{equation*} \mathcal{J}^- := \left\{ T \in B(\Hil) \vmid \exists B \in \mathcal{J}, \forall n\in \mathbb{N},\ \sum_{j=1}^n s_j(T) \le \sum_{j=1}^n s_j(B) \right\}. \end{equation*}An ideal \(\mathcal{J}\) is said to be arithmetic mean closed if \(\mathcal{J} = \mathcal{J}^-\text{.}\) Common examples of arithmetic mean closed ideals are the Schatten ideals \(\mathcal{C}_p\) of which the trace-class ideal \(\mathcal{C}_1\) and Hilbert–Schmidt ideal \(\mathcal{C}_2\) are special cases.
In [15], Kaftal and Weiss investigated the relationship between an ideal \(\mathcal{J}\) and the elements of its image \(E(\mathcal{J})\) under a trace-preserving conditional expectation onto an atomic masa \(\mathcal{A}\text{,}\) and they established the following characterization ([15] Corollary 4.4).
For every operator ideal \(\mathcal{J}\text{,}\) \(E(\mathcal{J}) = \mathcal{J}^- \cap \mathcal{A}\text{.}\)
Our next result says if an operator \(N\) can be diagonalized by a unitary \(U=I+K\) with \(K \in \mathcal{J}\) then the diagonals of \(N\) and its diagonalization differ by an element of the arithmetic mean closure of \(\mathcal{J}^2\text{.}\)
Let \(N\) be a diagonal operator, \(\mathcal{J}\) an operator ideal, and \(U = I+K\) a unitary with \(K \in \mathcal{J}\text{.}\) Then \(E(UNU^{*}-N) \in (\mathcal{J}^2)^-\text{.}\)
Irrespective of the condition \(K \in \mathcal{J}\text{,}\) note that \(U = I+K\) is unitary if and only if \(K\) is normal and \(K+K^{*} = -K^{*}K\) because
\begin{align*} UU^{*} &= I + K + K^{*} + KK^{*}\\ U^{*}U &= I + K + K^{*} + K^{*}K. \end{align*}Then
\begin{align*} E(UNU^{*}-N) &= E(KN+NK^{*}+KNK^{*})\\ &= E(KN)+E(NK^{*}) + E(KNK^{*})\\ &= E(K)N+NE(K^{*}) + E(KNK^{*})\\ &= E(K+K^{*})N + E(KNK^{*}) \in (\mathcal{J}^2)^-, \end{align*}by Corollary 3.6.
When \(\mathcal{J} = \mathcal{C}_2\text{,}\) which is the primary concern in this paper, we can say more.
Suppose \(N\) is a normal operator. There is an atomic masa such that for every unitary \(U = I + K\) with \(K\) Hilbert–Schmidt, \(E(UNU^{*}-N)\) is trace-class and has trace zero. Moreover, if \(N\) is diagonalizable, any atomic masa containing \(N\) suffices.
Suppose first that \(N\) is diagonalizable and consider an atomic masa in which \(N\) lies. Let \(U = I+K\) be unitary with \(K\) Hilbert–Schmidt. By Proposition 3.7 with \(\mathcal{J} = \mathcal{C}_2\) and its proof, each term of \(E(UNU^{*}-N) = E(K+K^{*})N + E(KNK^{*})\) is trace-class because \(K+K^{*} = -K^{*}K\) and \(KNK^{*}\) are trace-class, and because the trace-class is arithmetic mean closed (in fact, it is the smallest arithmetic mean closed ideal). Then, because the conditional expectation is trace-preserving, we find
\begin{align*} \trace\big(E(KNK^{*})\big) &= \trace(KNK^{*}) = \trace(K^{*}KN)\\ &= -\trace((K+K^{*})N) = -\trace(E(K+K^{*})N), \end{align*}and therefore \(\trace\big(E(UNU^{*}-N)\big) = 0\text{.}\)
Now suppose \(N\) is an arbitrary normal operator. By Voiculescu's extension [16] of the Weyl–von Neumann–Berg theorem we can write \(N = D+J\) where \(D\) is diagonalizable and \(J\) is Hilbert–Schmidt. Then \(UJU^{*}-J = KJ+JK^{*}+KJK^{*}\) and each term is trace-class. Moreover,
\begin{align*} \trace(KJK^{*}) &= \trace(K^{*}KJ) = -\trace((K+K^{*})J)\\ &= -\trace(KJ)-\trace(K^{*}J) = -\trace(KJ)-\trace(JK^{*}), \end{align*}and hence \(\trace(UJU^{*}-J) = 0\text{.}\) Therefore, if \(E\) is a conditional expectation onto an atomic masa containing \(D\text{,}\) then \(E(UNU^{*}-N) = E(UDU^{*}-D) + E(UJU^{*}-J)\) has trace zero.
The previous theorem establishes a kind of trace invariance property for arbitrary normal operators. To see why we use this terminology, consider that a trace-class operator \(A\) has a trace which is invariant under unitary conjugation. That is, for any unitary \(U\text{,}\) \(\trace A = \trace (UAU^{*})\text{.}\) Rearranging, we can write this as \(\trace(UAU^{*}-A) = 0\text{,}\) and since the canonical expectation is trace-invariant, we can rewrite this as \(\trace\big(E(UAU^{*}-A)\big) = 0\text{.}\) Under more restrictive hypotheses, Theorem 3.8 ensures the same condition for normal operators instead of trace-class operators.
The reader may have noticed that the normality in the previous theorem was only used in order to write the operator as a Hilbert–Schmidt perturbation of a diagonal operator. Therefore, the above theorem remains valid under this substitution of the hypothesis, and a slightly more general result is obtained.
One may wonder if in Proposition 3.7 and Theorem 3.8 we may take any trace-preserving conditional expectation instead of the special ones chosen. The answer is negative in general as this example shows. Consider commuting positive operators \(C,S\) in \(B(\Hil)\) with zero kernel satisfying \(C^2 + S^2 = I\text{.}\) Then consider the operators \(P,U \in M_2\big(B(\Hil)\big) \cong B(\Hil \oplus \Hil)\)
\begin{equation*} P := \frac{1}{\sqrt{2}} \begin{pmatrix} I & I \\ I & I \\ \end{pmatrix} \qquad U := \begin{pmatrix} C & S \\ -S & C \\ \end{pmatrix}, \end{equation*}which are a projection and a unitary, respectively. Thus
\begin{equation*} UPU^{*} = \frac{1}{\sqrt{2}} \begin{pmatrix} I+2CS & C^2 - S^2 \\ C^2 - S^2 & I-2CS \\ \end{pmatrix} \end{equation*}Now, choose \(S = \diag(\sin(\theta_n))\) and \(C = \diag(\cos(\theta_n))\) with \((\theta_n) \in \ell^2 \setminus \ell^1\text{.}\) Then \(S \in \mathcal{C}_2, C-I \in \mathcal{C}_1\) and hence \(U-(I \oplus I) \in \mathcal{C}_2\text{.}\) Moreover, \(2CS = \diag(\sin(2\theta_n))\) which is Hilbert–Schmidt but not trace-class. Thus, if \(E\) is the expectation onto an atomic masa containing \(C,S\text{,}\) and \(\tilde{E} := E \oplus E\text{,}\) then \(\tilde{E}(UPU^{*}-P) = \frac{1}{\sqrt{2}} (2CS \oplus -2CS) \in \mathcal{C}_2 \setminus \mathcal{C}_1\text{.}\)