MATHEMATICS

V. M. BRODSKII

ON THE TRIANGULAR REPRESENTATION OF OPERATORS CLOSE TO UNITARY ONES, AND THE MULTIPLICATIVE FACTORIZATION OF THEIR CHARACTERISTIC FUNCTIONS

(Presented by Academician A. Yu. Ishlinskii, 19 V 1969)

Let $\mathfrak H$ be a separable Hilbert space, $\mathfrak R$ the ring of bounded linear operators acting in $\mathfrak H$, and $\mathfrak S_\omega$ a symmetrically normed (s.n.) ideal of the ring $\mathfrak R$, introduced by V. I. Matsaev ($^1$).

I. Ts. Gohberg and M. G. Krein in the paper ($^2$) obtained a triangular representation of operators $T \in \mathfrak R$ having unitary spectrum. In the same paper, as well as in the work ($^3$), multiplicative factorizations of the characteristic functions of these operators are given. The results listed are extended in the present paper to operators $T \in \mathfrak R$ which, generally speaking, do not have unitary spectrum.

Lemma 1. If an invertible operator $T \in \mathfrak R$ possesses a discrete $$ chain $\mathfrak P={P_j}0^\infty$, then there exists an operator $V \in \mathfrak R$ possessing the chain $\mathfrak P$ and satisfying the conditions $\Delta P_j V \Delta P_j=0$ $(\Delta P_j=P_j-P,\ j=1,2,\ldots)$, such that*

[
T=D(I+V)\left(D=\int_{(\mathfrak P)} e^{i\varphi(P)}\,dP\right),
]

where the integral converges strongly, and the scalar function $\varphi(P)$ is determined from the relations $e^{i\varphi(P_j)}\Delta P_j=\Delta P_jT\Delta P_j$. If, in addition, $I-T^T \in \mathfrak S_\omega$, then the operator $V$ is Volterra.*

Proof. The strong convergence of the integral $D$ was proved in the paper ($^4$). Let $g_j$ $(j=1,2,\ldots)$ be a unit vector of the one-dimensional subspace $\Delta P_j \mathfrak H$. Since

[
Tg_j=\beta_{j,1}g_1+\ldots+\beta_{j,j-1}g_{j-1}+e^{i\varphi(P_j)}g_j,
]

the numbers $e^{i\varphi(P_j)}$ belong to the spectrum of the operator $T$. By the invertibility of $T$,

[
\inf_j |e^{i\varphi(P_j)}|>0,
]

and, consequently, the operator $D$ is invertible.

Put $V=D^{-1}T-I$. Then $T=D(I+V)$ and

[
VP_j=D^{-1}P_jTP_j-P_j=P_j(D^{-1}T-I)P_j=P_jVP_j,
]

[
\Delta P_jV\Delta P_j=\Delta P_jD^{-1}\Delta P_jT\Delta P_j-\Delta P_j=0.
]

Let us note that

[
\Delta P_j(I+V^)^{-1}\Delta P_j=\Delta P_j,\qquad
\Delta P_jD^D(I+V)\Delta P_j=\Delta P_jD^*D\Delta P_j.
]

From the equalities

[
I-D^D=\sum_{j=1}^{\infty}\Delta P_j(I-D^D)\Delta P_j =
]

[
=\sum_{j=1}^{\infty}\Delta P_j\bigl((I+V^)^{-1}-D^D(I+V)\bigr)\Delta P_j =
]

[
=\sum_{j=1}^{\infty}\Delta P_j(I+V^)^{-1}H\Delta P_j
\qquad (H=I-T^T)
]

* For terminology and notation see ($^4,^5$).

it follows (see also (⁶), p. 109) that (I-D^{*}D\in\mathfrak{S}_{\omega}). Hence one can obtain the relation

[
S=(I+V^{})^{-1}(I-(I+V^{})(I+V))\in\mathfrak{S}_{\omega}.
]

Moreover, (\Delta P_j S \Delta P_j=0), and therefore, by (¹), the integral

[
\int_{[\mathfrak{P}]} PS\,dP
=
\sum_{j=1}^{\infty} P_{j-1}(I+V^{*})^{-1}\Delta P_j
-
\sum_{j=1}^{\infty} P_{j-1}(I+V)\Delta P_j
=
]

[

-\sum_{j=1}^{\infty} P_{j-1}V\Delta P_j

-V
]

converges uniformly and is a Volterra operator. The lemma is proved.

Let (T\in\mathfrak{R}) and (I-T^{}T\in\mathfrak{S}{\omega}). Denote by (\mathfrak{H}_0) the closure of the linear span of all root subspaces of the operator (T) corresponding to its eigenvalues not lying on the unit circle. Introduce the orthoprojectors (P^{(0)}) and (P^{(1)}) respectively onto (\mathfrak{H}_0) and (\mathfrak{H}_1=\mathfrak{H}\ominus\mathfrak{H}_0). Consider the operator (T_0), induced by the operator (P^{(0)}TP^{(0)}) in (\mathfrak{H}_0), and the operator (T_1), induced by the operator (P^{(1)}TP^{(1)}) in (\mathfrak{H}_1). The operator (T_0) has a discrete chain (\mathfrak{P}^{(0)}={Q_j})}^{\infty, while the operator (T_1)** has a maximal chain (\mathfrak{P}^{(1)}={Q}), separating its spectrum (²). We shall also need the maximal chain (\mathfrak{P}={P}), which is the union of the chains (\mathfrak{P}_0={QP^{(0)}}) ((Q\in\mathfrak{P}^{(0)})) and (\mathfrak{P}_1={P^{(0)}+QP^{(1)}}) ((Q\in\mathfrak{P}^{(1)})), and the scalar function (\varphi(P)) ((P\in\mathfrak{P})), defined by the following conditions:

I. (e^{i\varphi(P_j)}\Delta P_j=\Delta P_jT\Delta P_j) ((\Delta P_j=P_j-P_{j-1},\; P_j\in\mathfrak{P}_0)).

II. (\varphi(P)) ((P\in\mathfrak{P}1)) coincides with the least of the numbers (\tau) for which the arc (e^{it}) ((0\leq t\leq\tau)) contains the spectrum of the operator (T_1)).}h=PT_1h) acting in the subspace (P\mathfrak{H}_1) ((h\in P\mathfrak{H

Theorem 1. If the operator (T\in\mathfrak{R}) is invertible and (I-T^{*}T\in\mathfrak{S}_{\omega}), then

[
T=\int_{[\mathfrak{P}]} e^{i\varphi(P)}\,dP
\left(I+\int_{[\mathfrak{P}]}(I-PHP)^{-1}PH\,dP\right)^{-1}
\quad (H=I-T^{*}T),
\tag{1}
]

where the first integral converges strongly, and the second uniformly.

Proof. Using Lemma 1 and Theorem 1 of paper (²), it is not difficult to represent the operator (T) in the form (T=D(I+V)), where

[
D=\int_{\mathfrak{P}} e^{i\varphi(P)}\,dP,
]

and (V) is a Volterra operator possessing the chain (\mathfrak{P}). From Theorems 3.1 and 7.1 of paper (⁵) follows the equality

[
(I+V)^{-1}
=
I+\int_{[\mathfrak{P}]}(I-PHP)^{-1}PH\,dP.
]

Theorem 1 shows that an invertible operator (T\in\mathfrak{R}) satisfying the condition (I-T^{*}T\in\mathfrak{S}_{\omega}) can be represented as the sum of a normal operator and a Volterra operator possessing one and the same maximal chain.

Let

[
T\in\mathfrak{R},\qquad
J_T=\operatorname{sign}(I-T^{}T),\qquad
\mathfrak{K}_T=\overline{(I-T^{}T)\mathfrak{H}}.
]

The function

[
\theta_T(\zeta)=
\left.(T-\zeta J_{T^{}}|I-TT^{}|^{1/2}(I-\zeta T^{})^{-1}|I-T^{}T|^{1/2})\right|_{\mathfrak{K}_T},
]

* For definiteness, we consider only the case where (\dim \mathfrak{H}_0=\infty).

** The spectrum of the operator (T_1) lies on the unit circle (⁶).

acting from the space (\mathfrak R_T) into the space (\mathfrak R_{T^*}), is called the characteristic function of the operator (T).

Lemma 2. If the operator (T \in \mathfrak R) is invertible and (I-T^*T \in \mathfrak S_\omega), then

[
\theta_T(\xi)=U_0\left(I-J_T|H|^{1/2}FP^{(1)}(D^D)^{-1/2}
(\xi D^+(D^D)^{1/2})(I-\xi T^)^{-1}|H|^{1/2}\right)
]
[
\times
\left(I-J_T|H|^{1/2}FP^{(0)}(D^D)^{-1/2}
(\xi D^+(D^D)^{1/2})(I-\xi T^)^{-1}|H|^{1/2}\right)\bigm|_{\mathfrak R_T}
]
[
(F=(I+(D^*D)^{1/2}(I+V))^{-1}).
]

The operator (U_0) satisfies the relation (U_0J_TU_0^=J_{T^}), maps the whole space (\mathfrak R_T) onto the whole space (\mathfrak R_{T^*}), and is computed by the formula

[
U_0=(T^)^{-1}(I-J_T|H|^{1/2}F^|H|^{1/2}).
]

Let (\mathfrak S) be an arbitrary s.n. ideal of the ring (\mathfrak R).

Lemma 3. If the operator (T \in \mathfrak R) is invertible and the operator (I-T^*T) belongs simultaneously to the s.n. ideals (\mathfrak S) and (\mathfrak S_\omega), then

[
I-J_T|H|^{1/2}FP^{(0)}(D^D)^{-1/2}
(\xi D^+(D^D)^{1/2})(I-\xi T^)^{-1}|H|^{1/2}
=
]
[
=\prod_{j=1}^{\infty}
\left(
Q^{(j)}\frac{\xi_j-\xi}{1-\xi\bar{\xi}_j}\frac{|\xi_j|}{\xi_j}P^{(j)}
\right),
\tag{2}
]

where the infinite product converges in the norm of (\mathfrak S), the numbers (\xi_j) ((j=1,2,\ldots)) are determined from the relations (\xi_j\Delta P_j=\Delta P_jT\Delta P_j), (P^{(j)}) ((j=1,2,\ldots)) are one-dimensional projectors computed by the formula

[
P^{(j)}=\frac{1+\xi_j}{1-\bar{\xi}_j}\,
J_T|H|^{1/2}F\Delta P_jF^*|H|^{1/2}
]

and (Q^{(j)}=I-P^{(j)}).

Lemma 4. If the operator (T \in \mathfrak R) is invertible and the operator (I-T^*T) belongs simultaneously to the s.n. ideals (\mathfrak S) and (\mathfrak S_\omega), then

[
I-J_T|H|^{1/2}FP^{(1)}(D^D)^{-1/2}
(\xi D^+(D^D)^{1/2})(I-\xi T^)^{-1}|H|^{1/2}
=
]
[
=
\int_{(\mathfrak B_1)}
\left(
I-\frac{e^{i\varphi(P)}+\xi}{e^{i\varphi(P)}-\xi}\,
d\left(2J_T|H|^{1/2}FP^{(1)}PP^{(1)}F^*|H|^{1/2}\right)
\right).
\tag{3}
]

The integral (3) converges in the norm of (\mathfrak S).

Lemma 4 for the special case when (|H|\le 1) and the spectrum of the operator (T) lies on the unit circle was given in paper (5).

Let the function (\theta(\xi)), whose values are linear bounded operators in (\mathfrak H), satisfy the following conditions: 1) (\theta(\xi)) is holomorphic in the domain obtained by deleting from the disk (|\xi|<1) at most a countable set of isolated points ({\xi_j}_{j=1}^q) ((q\le\infty,\ \xi_j\ne0)); 2) (\theta(\xi)) is invertible at least at one point (\xi) ((|\xi|<1)); 3) (|\theta(\xi)|\le1) ((|\xi|<1)) and (|\theta(0)f|<|f|) for all (f\ne0).

Then, as shown in article (7), there exists a contraction (T) such that (\theta(\xi)=U_\theta_T(\xi)U), where the operators (U) and (U_) isometrically map the spaces (\mathfrak H) and (\mathfrak R_{T^}), respectively, onto (\mathfrak R_T) and (\mathfrak H). If, in addition: 4) there exists a point (\xi_0) ((|\xi_0|<1)) such that the operator (I-\theta^(\xi_0)\theta(\xi_0)) belongs simultaneously to the ideals (\mathfrak S) and (\mathfrak S_\omega), then (see (2)) the operator (I-T^*T) belongs to these ideals. Hence from Lemmas 2, 3, and 4 it follows that

Theorem 2. If the function (\theta(\xi)), whose values are linear bounded operators in (\mathfrak H), satisfies conditions 1)—4), then

[
\theta(\xi)=U_{(\omega)}
\int_{(\mathfrak B_1)}
\left(
I-\frac{e^{i\varphi(P)}+\xi}{e^{i\varphi(P)}-\xi}\,dF(P)
\right)
\prod_{j=1}^{q}
\left(
Q^{(j)}+\frac{\xi_j-\xi}{1-\xi\bar{\xi}_j}\frac{|\xi_j|}{\xi_j}P^{(j)}
\right)U,
\tag{4}
]

where $\mathfrak{P}1$ is a maximal chain in some space $\mathfrak{H}_1 \subset \mathfrak{H}$; $\varphi(P)$ $(P \in \mathfrak{P}_1,\ 0 \leq \varphi(P) \leq 2\pi)$ is a nondecreasing scalar function; $F(P)$ $(P \in \mathfrak{P}_1)$ is a positive operator-valued function; $P^{(j)}$ $(j=1,2,\ldots)$ are one-dimensional orthoprojections; $Q^{(j)} = I - P^{(j)}$, and $U$.}$ is an isometric operator mapping the space $\mathfrak{H}_T$ onto $\mathfrak{H}$. The integral and the infinite product converge in the norm of $\mathfrak{S

Theorem 2 generalizes results of V. P. Potapov (⁸) and Yu. P. Ginzburg (⁹, ¹⁰)* on the multiplicative representation of contractive analytic operator-valued functions.

Odessa Institute of National Economy

Received
8 IV 1969

REFERENCES

¹ V. I. Matsaev, DAN, 139, No. 3, 548 (1961).
² I. Ts. Gokhberg, M. G. Krein, DAN, 164, No. 4, 731 (1965).
³ V. M. Brodskii, DAN, 173, No. 2, 256 (1967).
⁴ V. M. Brodskii, M. S. Brodskii, DAN, 181, No. 3 (1968).
⁵ I. Ts. Gokhberg, M. G. Krein, Acta Sci. Math., 25, 1–2, 90 (1964).
⁶ I. Ts. Gokhberg, M. G. Krein, Introduction to the Theory of Linear Non-Selfadjoint Operators, “Nauka,” 1965.
⁷ B. Sz.-Nagy, C. Foias, Acta Sci. Math., 25, 1–2, 37 (1964).
⁸ V. P. Potapov, Tr. Mosk. matem. obshch., 4, 125 (1955).
⁹ Yu. P. Ginzburg, DAN, 117, 171 (1957).
¹⁰ Yu. P. Ginzburg, DAN, 170, No. 1, 23 (1966).
¹¹ Yu. P. Ginzburg, UMN, 22, 1 (133), 163 (1967).

* In the paper (¹¹), Yu. P. Ginzburg obtained a multiplicative representation of functions not contained in the class of functions considered in Theorem 2.