MATHEMATICS

M. G. KREIN

ON PERTURBATION DETERMINANTS AND THE TRACE FORMULA FOR UNITARY AND SELF-ADJOINT OPERATORS

(Presented by Academician V. I. Smirnov on 5 I 1962)

1. In what follows \(\mathfrak H\) denotes a separable Hilbert space, \(\mathfrak R\) the linear ring of all bounded operators acting in \(\mathfrak H\), and \(\mathfrak S\) the two-sided ideal in \(\mathfrak R\) consisting of all nuclear operators, i.e., of all completely continuous operators \(A\) such that
\[ \operatorname{Sp}|A|=\operatorname{Sp}(A^*A)^{1/2}<\infty. \]
For \(A\in\mathfrak S\) the two functionals \(\operatorname{Sp} A\) and \(\det(I+A)\) are meaningful.

If \(G\) is a domain in the complex plane, \(A_z\) \((z\in G)\) is a holomorphic operator-valued function with values in \(\mathfrak S\), and \(\Delta(z)=\det(I+A_z)\), then at any point \(z\in G\) for which \(\Delta(z)\ne0\) the formula
\[ \Delta'(z)/\Delta(z)=\operatorname{Sp}\bigl[(I+A_z)^{-1}(dA_z/dz)\bigr] \]
holds.

Let \(A\) be a linear operator with domain of definition \(\mathfrak D(A)\subset\mathfrak H\) and with nonempty set \(\rho(A)\) of all regular points (i.e., all complex points \(z\) for which there exists a resolvent \(R_z(A)=(A-zI)^{-1}\in\mathfrak R\)). If \(B\) is another linear operator with \(\mathfrak D(B)=\mathfrak D(A)\) and, at least at one point \(z\in\rho(A)\), the condition
\[ (B-A)R_z(A)\in\mathfrak S \]
is satisfied, then this condition will be satisfied for all \(z\in\rho(A)\), and for these \(z\) the perturbation determinant is meaningful:
\[ \Delta_{B/A}(z)=\det\bigl[(B-zI)(A-zI)^{-1}\bigr] =\det\bigl[I+(B-A)R_z(A)\bigr], \tag{1} \]
which is a holomorphic function on each connected component of \(\rho(A)\). From formula (1) it follows that at points \(z\in\rho(A)\) for which \(\Delta_{B/A}(z)\ne0\):
\[ \frac{d}{dz}\Delta_{B/A}(z)/\Delta_{B/A}(z) =\operatorname{Sp}\bigl[R_z(A)-R_z(B)\bigr] \quad \text{for } z\in\rho(A)\cap\rho(B). \tag{2} \]

Perturbation determinants for various pairs of operators \(A,B\) have been used in various works of the author \((^{1,2})\). Apparently, without knowing of these works, Sh. T. Kuroda \((^3)\) came to the consideration of perturbation determinants. He considered the properties of \(\Delta_{B/A}(z)\) for the case of two closed linear operators \(A,B\), for which \(\mathfrak D(A)=\mathfrak D(B)\) is dense in \(\mathfrak H\). In this case one may assert \((^4)\) that if certain connected components \(G(A)\subset\tilde\rho(A)\) and \(G(B)\subset\tilde\rho(B)\) have a common point, then \(G(A)=G(B)\), and on \(G(A)\) the function \(\Delta_{B/A}(z)\) is a meromorphic function, where the order of \(\Delta_{B/A}(z)\) at any point \(z\in G(A)\) is computed in a natural way \((^{1,3})\). Here \(\tilde\rho(A)\) \((\tilde\rho(B))\) denotes the set of complex points consisting of \(\rho(A)\) \((\rho(B))\) and all isolated points of the spectrum of the operator \(A\) \((B)\) that are eigenvalues of \(A\) \((B)\) of finite algebraic multiplicity.

In a plenary lecture at the IV All-Union Mathematical Congress (6 VII 1961) the author indicated that in known cases the notion of a generalized perturbation determinant \(\widetilde\Delta_{B/A}(z)\) may be introduced. For example, this is possible when the operators \(A\) and \(B\) are resolvent comparable, i.e., when the set \(\rho=\rho(A)\cap\rho(B)\) is nonempty and, at least for one point \(z\in\rho\) (and then for all \(z\in\rho\)),
\[ R_z(B)-R_z(A)\in\mathfrak S. \]
In this case, choosing

arbitrary \(z_0 \in \rho\), one may set \(\widetilde{\Delta}_{B/A}(z)=\Delta_{B_0/A_0}(z)\), where \(A_0=R_{z_0}(A)\), \(B_0=R_{z_0}(B)\).

It is not difficult to show that, up to a multiplicative constant, the determinant \(\Delta_{B/A}(z)\) does not depend on the choice of the point \(z_0 \in \rho\); for \(z \in \rho\), (2) will again hold with \(\Delta\) replaced by \(\widetilde{\Delta}\).

1. All assertions of Theorem 1 were established already in \((^2)\), except for the last of them, which contains a positive answer to the question posed in \((^2)\) (p. 618).

Theorem 1. Let \(H_1\) be a self-adjoint operator, \(V \in \mathfrak{S}\), \(H_2=H_1+V\), and \(\Delta(z)=\Delta_{H_2/H_1}(z)\). Then, under the corresponding definition of \(\ln \Delta(z)\),

\[ \ln \Delta(z)=\int_{-\infty}^{\infty} \frac{\xi(\lambda)}{\lambda-z}\,d\lambda \qquad (\operatorname{Im} z \ne 0), \tag{3} \]

where \(\xi(\lambda)\) is a real measurable function, and moreover

\[ 1)\ \int_{-\infty}^{\infty} |\xi(\lambda)|\,d\lambda \leq \operatorname{Sp}|V|;\qquad 2)\ \int_{-\infty}^{\infty} \xi(\lambda)\,d\lambda=\operatorname{Sp} V. \tag{4} \]

If the operator \(V\) has in all \(p\) positive (in all \(q\) negative) eigenvalues, then almost everywhere \(\xi(\lambda)\leq p\) \((\xi(\lambda)\geq -q)\). In particular, if the operator \(V\) is nonnegative, then almost everywhere \(\xi(\lambda)\geq 0\). For any complex-valued continuously differentiable function \(\Phi(\lambda)\) \((-\infty<\lambda<\infty)\) such that

\[ \Phi(\lambda)=\int_{-\infty}^{\infty} e^{i\lambda t}\,d\omega(t),\qquad \int_{-\infty}^{\infty} |t|\,|d\omega(t)|<\infty, \tag{5} \]

one always has \(\Phi(H_2)-\Phi(H_1)\in\mathfrak{S}\), and the trace formula is valid

\[ \operatorname{Sp}\,[\Phi(H_2)-\Phi(H_1)] = \int_{-\infty}^{\infty} \xi(\lambda)\Phi'(\lambda)\,d\lambda. \tag{6} \]

Formula (3) was first discovered by I. M. Lifshits \((^5)\) in connection with certain questions in the quantum theory of crystals (see also his survey \((^6)\)). In deriving this formula, the author mentioned assumed that \(V\) is a finite-dimensional operator, and imposed a number of smoothness requirements on the spectral function \(E(\lambda)\) of the operator \(H_1\) (moreover, when a discrete spectrum appeared in \(H_2\), I. M. Lifshits wrote formula (3) in another form).

1. All assertions of Theorem 1 have their analogues in the theory of unitary operators. In particular, the following holds:

Theorem 2. Let \(U_1\) and \(U_2\) be two unitary operators, with \(U_2-U_1\in\mathfrak{S}\). Then, up to an additive constant, there exists a unique real function \(\eta(t)\in L_1(-\pi,\pi)\) such that

\[ \operatorname{Sp}\,[\Psi(U_2)-\Psi(U_1)] = \int_{-\pi}^{\pi} \eta(t)\,d\Psi(e^{it}), \tag{7} \]

where \(\Psi(\xi)\) \((|\xi|=1)\) is any function admitting an expansion

\[ \Psi(\xi)=\sum_{n=-\infty}^{\infty} c_n \xi^n,\qquad \sum_{n=-\infty}^{\infty} |n c_n|<\infty. \tag{8} \]

The function \(\eta(t)\) can be obtained from the formula

\[ \eta(t)=\frac{1}{\pi}\lim_{\rho\uparrow 1}\arg \Delta_{U_2/U_1}(\rho e^{it})+\mathrm{const}\quad \text{(almost everywhere).} \tag{9} \]

We shall give some explanations concerning the proof of the theorem. If \(U_2-U_1\in\mathfrak{S}\), then \(U_2=U_1(I+T)\), where \(T\in\mathfrak{S}\). Since \(I+T\) is a unitary operator, it follows that \(T=\sum t_j P_j\), where \(P_j=(\,\cdot\,,\varphi_j)\varphi_j\) \((j\in J)\) is a certain finite or infi-

an infinite sequence of pairwise orthogonal projectors, and \(1+\tau_j=\exp(i\theta_j)\) \((-\pi<\theta_j\leqslant \pi)\) are points of the unit circle, with \(\operatorname{Sp}|T|=\sum |\tau_j|<\infty\). Thus,

\[ U_2=U_1\left(I+\sum_{j\in J}\tau_j P_j\right) =U_1\prod_{j\in J}(I+\tau_jP_j), \]

\[ \operatorname{Sp}\left|\ln_0(U_1^{-1}U_2)\right| =\sum_{j\in J}|\theta_j|<\infty, \]

where by \(\ln_0(U_1^{-1}U_2)\) we denote that operator value of \(\ln(U_1^{-1}U_2)\) whose spectrum lies in the interval \((-\pi i,\pi i]\), open on the left.

Let us first consider the case of a one-dimensional perturbation, i.e., when
\(U_2=U_1(I+\tau(\,\cdot\,,\varphi)\varphi)\), where \(\|\varphi\|=1\),
\(1+\tau=\exp(i\theta)\) \((-\pi<\theta<\pi)\). In this case it is easily shown that

\[ \Delta_{U_2/U_1}(z) = e^{i\frac{\theta}{2}} \left[ \cos\frac{\theta}{2} + i\sin\frac{\theta}{2} \int_{-\pi}^{\pi} \frac{e^{it}+z}{e^{it}-z}\,d(E(t)\varphi,\varphi) \right] \quad (|z|\ne 1), \]

where \(E(t)\) is the spectral function of the operator \(U_1\). Hence it is already not difficult to conclude that

\[ \ln \Delta_{U_2/U_1}(z) = i\frac{\theta}{2} + i\int_{-\pi}^{\pi} \frac{e^{it}+z}{e^{it}-z}\eta_1(t)\,dt, \qquad \int_{-\pi}^{\pi}\eta_1(t)\,dt=\frac{\theta}{2}, \]

where \(|\eta_1(t)|\leqslant 1\) and \(\operatorname{sign}\theta\cdot \eta_1(t)\geqslant 0\)
\((-\pi\leqslant t\leqslant \pi)\). Recalling relation (2), we verify for the case under consideration the validity of (7) with
\(\eta(t)=\eta_1(t)\) and \(\Psi(\zeta)=\Psi_z(\zeta)=(\zeta-z)^{-1}\) \((|z|\ne1)\). Since the trace formula has been obtained for the family of functions \(\Psi_z(\zeta)\) \((|z|\ne1)\), its extension to the entire class of functions \(\Psi(\zeta)\) indicated in Theorem 2 presents no particular difficulty.

In the case of the general relation (9), the function \(\eta(t)\) in (7) can be obtained (cf. (2), § 4) in the form of a sum \(\sum \eta_j(t)\) convergent in the \(L_1(-\pi,\pi)\) metric, where \(\eta_j(t)\) \((j=1,2,\ldots)\), obtained in the manner indicated above, is the function from the trace formula for the unitary operators \(U^{(j)}\) and \(U^{(j+1)}\), with

\[ U^{(j+1)}=U^{(j)}(I+\tau_jP_j) \quad (j=1,2,\ldots;\ U^{(1)}=U_1). \]

This way of establishing the trace formula (7) is also interesting because it supplies a function \(\eta(t)\) possessing the following two properties:

\[ \int_{-\pi}^{\pi}|\eta(t)|\,dt \leqslant \operatorname{Sp}\left|\ln_0(U_1^{-1}U_2)\right|, \qquad i\int_{-\pi}^{\pi}\eta(t)\,dt = \operatorname{Sp}\ln_0(U_1^{-1}U_2). \tag{10} \]

The last equality uniquely determines the function \(\eta(t)\) in the trace formula (7). This equality may be regarded as the requirement that formula (7) be valid for the function \(\Psi(\zeta)=\ln_0\zeta\), if in doing so the expression
\(\operatorname{Sp}[\ln_0 U_2-\ln_0 U_1]\) is understood as
\(\operatorname{Sp}[\ln_0(U_1^{-1}U_2)]\). The function \(\eta(t)\), normalized by equality (10), will be called the spectral shift function of the ordered pair of unitary operators \(U_1,U_2\). From its construction it also follows that, if the operator \(U_1^{-1}U_2\) has in all \(p\) eigenvalues in the closed half-plane \(\operatorname{Im} z\geqslant 0\) (in all \(q\) eigenvalues in the open half-plane \(\operatorname{Im} z<0\)), then almost everywhere
\(\eta(t)\leqslant p\) \((\eta(t)\geqslant -q)\).

1. It is easy to see that two self-adjoint operators \(H_1\) and \(H_2\) are resolvent comparable if and only if the difference of their Cayley transforms
   \(U_k=(iI-H_k)(iI+H_k)^{-1}\) \((k=1,2)\) is nuclear:
   \(U_2-U_1\in\mathfrak{S}\).

From Theorem 2 and the other assertions concerning the function \(\eta(t)\) there follows

Theorem 3. To every ordered pair of resolvent-comparable self-adjoint operators \(H_1,H_2\) there corresponds, up to an additive constant,

of the summand a unique real measurable function \(\xi(\lambda)\) \((-\infty<\lambda<\infty)\) such that \((1+\lambda^2)^{-1}\xi(\lambda)\in L_1(-\infty,\infty)\) and

\[ \operatorname{Sp}\left[(H_1-zI)^{-1}-(H_2-zI)^{-1}\right] = \int_{-\infty}^{\infty}\frac{\xi(\lambda)}{(\lambda-z)^2}\,d\lambda \quad (z\in\rho(H_1)\cap\rho(H_2)). \tag{11} \]

If one sets \(\Delta(z)=\widetilde{\Delta}_{H_2/H_1}(z)\) and chooses a single-valued harmonic branch \(\arg\Delta(z)\) \((\operatorname{Im}z>0)\), then the required function \(\xi(\lambda)\) can be obtained from the formula

\[ \xi(\lambda)=\frac{1}{\pi}\lim_{\varepsilon\downarrow 0}\arg\Delta(\lambda+i\varepsilon)+\mathrm{const} \quad \text{(almost everywhere).} \]

The function \(\xi(\lambda)\) will be semibounded from below, provided the following conditions are satisfied: 1) \(\mathfrak{D}(H_1)=\mathfrak{D}(H_2)\); 2) the form \(((H_2-H_1)f,f)\) \((f\in\mathfrak{D}(H_1))\) has a finite number of negative squares; and 3) there exist constants \(\alpha,\beta\) \((0\le \alpha<1,\ \beta\ge 0)\) such that
\[ \|(H_2-H_1)f\|\le \alpha\|H_1f\|+\beta\|f\| \]
for all \(f\in\mathfrak{D}(H_1)\).

In proving the last assertion one has to invoke a lemma which, as the author has found, is useful in various questions of perturbation theory.

If the spectra of the unitary operators \(U_k\) \((k=1,2)\) lie respectively on the arcs \(\zeta=\exp(i\theta)\) \((\alpha_k\le\theta\le\beta_k;\ k=1,2)\), then the spectrum of their product \(U_1U_2\) lies on the arc \(\zeta=\exp(i\theta)\) \((\alpha_1+\alpha_2\le\theta\le\beta_1+\beta_2)\).

We note that the totality of conditions 1) and 3) was used for another purpose in the paper \((^7)\).

Of course, the function \(\xi(\lambda)\), defined by (11), makes it possible to write the trace formula (6), which will be valid for the corresponding class of functions \(\Phi(\lambda)\).

1. If \(H_1\) is a semibounded operator (or, more generally, an operator having a certain spectral gap), then every self-adjoint operator \(H_2\) resolvent-comparable with \(H_1\) will also be such. In these cases the first two assertions of Theorem 3 can easily be obtained from Theorem 1 by passing from the operators \(H_1\) and \(H_2\) to their resolvents \(R_a(H_1)\) and \(R_a(H_2)\), where \(a\) is some real point regular for \(H_1\) and \(H_2\). Moreover, for the case when, for example, both operators \(H_1\) and \(H_2\) are semibounded from below, the trace formula for them can be obtained (on the basis of Theorem 3) under the condition
   \[ R_a^\nu(H_2)-R_a^\nu(H_1)\in\mathfrak{S}, \]
   where \(\nu>0\) and \(a\) is some* real point lying to the left of the spectra of \(H_1\) and \(H_2\) (concerning this condition, see \((^8,^9)\)).

Odessa Civil Engineering Institute

Received
1 XI 1961

REFERENCES CITED

\(^1\) M. G. Krein, UMN, 14, no. 3 (87) (1959); DAN, 130, No. 2 (1960).
\(^2\) M. G. Krein, Matem. sborn., 33 (75), 3 (1953).
\(^3\) S. T. Kuroda, Sci. Papers, Coll. Gen. Educ. 11, No. 1 (1961).
\(^4\) Ts. I. Gohberg, M. G. Krein, UMN, 12, No. 2 (74) (1957).
\(^5\) I. M. Lifshits, UMN, 7, No. 1 (1952).
\(^6\) M. Lifšic, Nuovo Cimento, No. 4 del Suppl., 3, ser. X (1956).
\(^7\) S. T. Kuroda, J. Math. Soc. Japan, 1, 11, No. 3 (1959).
\(^8\) M. Sh. Birman, DAN, 137, No. 4 (1961).
\(^9\) M. Sh. Birman, DAN, 143, No. 3 (1962).

* The validity of the condition does not depend on the choice of the point \(a\), and its strength decreases as \(\nu\) increases.