Abstract Generated abstract
This note studies the reduction of quasiunitary operators in Hilbert space to triangular form, under finite-dimensional nonunitarity and a regularity condition inside the unit disk. It defines a normalized characteristic matrix function for such operators and uses Potapov’s factorization theorem to construct an explicit triangular model with prescribed rank and signature of nonunitarity. The main result states that every operator in the considered class is unitarily equivalent, modulo its supplementary component, to this triangular model, whose spectrum and characteristic function are described. For quasiunitary contractions, the model is also used to give a criterion for completeness of eigenfunctions and associated functions in terms of an inequality involving the determinant of a finite matrix and the product of eigenvalues.
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MATHEMATICS
V. T. POLYATSKII
ON THE REDUCTION OF QUASIUNITARY OPERATORS TO TRIANGULAR FORM
(Presented by Academician A. N. Kolmogorov, 15 X 1956)
1. It is known that every matrix of finite order can be reduced, by means of a unitary transformation, to triangular form. In the paper (¹) an analogous question was solved for operators of class \((i\Omega)\). In the present note the problem of reducing to triangular form quasiunitary operators defined in a Hilbert space \(H\) is solved.
Definition. A linear operator \(T\), defined in a Hilbert space \(H\), is called quasiunitary if the operators \(I - T^*T\) and \(I - TT^*\) are finite-dimensional.
Here we shall consider quasiunitary operators \(T\) satisfying the conditions:
1) \(\operatorname{Dim}(I - T^*T) = \operatorname{Dim}(I - TT^*)\);
2) there exists at least one point \(\zeta_0\), \(|\zeta_0| < 1\), regular for the resolvent of the operator \(T\).
Denoting \(D_T = (I - T^*T)H\); \(D'_T = (I - TT^*)\), we shall call the dimension of \(D_T\) and \(D'_T\) the rank of nonunitarity of the operator \(T\), and the pair of numbers \((p, q)\), where \(p\) is the number of positive and \(q\) the number of negative squares of the form \(((I - T^*T)f, f)\) \((f \in H)\), the signature of nonunitarity of the operator \(T\).
The subspace \(G_T = H \ominus D_T\) is the largest subspace on which \(T^* = T^{-1}\).
The maximal subspace \(\mathfrak M_T \subseteq G_T\) invariant with respect to \(T\) will be called the additional component of the operator \(T\). If \(\mathfrak M_T = 0\), then the quasiunitary operator will be called simple.
Without loss of generality, one may assume that the point \(\zeta = 0\) does not belong to the spectrum of the operator \(T\).
It is not hard to show that the additional component coincides with the orthogonal complement of the linear closed span of \(T^{*k}D_T\) \((k = 0, \pm 1, \pm 2, \ldots)\).
2. The operator \(|I - TT^*|^{-1/2}\) has meaning on \(D'_T\); therefore, in view of the relations
\[ (T - \zeta I)(I - \zeta T^*)^{-1}|I - T^*T|^{1/2} = [T - \zeta(I - TT^*)(I - \zeta T^*)^{-1}]\,|I - T^*T|^{1/2}, \]
\[ TD_T \subseteq D'_T, \]
the operator function
\[ W(\zeta) = |I - TT^*|^{1/2}(T - \zeta I)(I - \zeta T^*)^{-1}|I - T^*T|^{1/2} \]
has meaning for those \(\zeta\) for which \(\overline{\zeta}^{-1}\) is a regular point of the operator \(T\).
\(W(\zeta)\) maps \(D_T\) into \(D'_T\) (⁴).
Considering orthonormal bases \(\{e_k\}\), \(\{e'_k\}\) \((k = 1, 2, \ldots, r)\) in \(D_T\) and \(D'_T\), respectively, we introduce the matrix function
\[ w_T(\zeta)=\left\|\bigl(W(\zeta)e_k,e'_j\bigr)\right\|_{k,j=1}^{r}, \]
which we shall call the normalized characteristic matrix-function of the quasiunitary operator \(T\). It is not difficult to show that this definition is equivalent to the definition of the analogous function in paper \((^{2})\); therefore all assertions of that paper are valid for \(w_T(\zeta)\).
Since \(w_T(\zeta)\) satisfies all the conditions of Potapov’s main theorem \((^{3})\), it can be represented in the following equivalent form:
\[ w_T(\zeta)=\prod_{k=1}^{\infty} \mathfrak U_k\left\{t(k)-\zeta p(k)\left[e^{i\varphi(k)}-\zeta t(k)\right]^{-1}p(k)J\right\}\mathfrak U_k^{-1}\times \]
\[ \times \int_{0}^{l} \exp\left[ \frac{\zeta+e^{i\varphi(x)}}{\zeta-e^{i\varphi(x)}}p^2(x)J\,dx \right]\mathfrak B, \tag{1} \]
where \(t(k)\) is a sequence of positive diagonal matrices having one and only one eigenvalue different from unity, and
\[ \prod_{k=1}^{\infty}\rho_k \]
converges \(\bigl(\rho_k\ne 1\) are the eigenvalues of the matrices \(t(k)\bigr)\); \(p^2(k)=|I-t^2(k)|\); \(\{\mathfrak U_k\}\) is a sequence of \(J\)-unitary matrices such that
\[ \prod_{k=1}^{\infty}\mathfrak U_k t(k)\mathfrak U_k^{-1} \]
converges. \(p(x)\) is a Hermitian nonnegative matrix-function satisfying the condition \(\operatorname{Sp} p^2(x)=1\) \((0\le x\le l)\); \(\varphi(k)\) and \(\varphi(x)\) are nondecreasing and bounded scalar functions, with \(\varphi(x+0)=\varphi(x)\).
- In this section we construct a triangular model \(T\) of a quasiunitary operator with given nonunitarity rank \(r\) and given signature \((p,q)\).
Let \(\mathscr H_r\) be a Euclidean \(r\)-dimensional space. Consider the spaces \(\mathbf H_1\) and \(\mathbf H_2\) of all vector-functions \(f(k)\) \((k=1,2,\ldots)\) and \(f(x)\) \((0\le x\le l)\) of discrete and continuous arguments \(k\) and \(x\), respectively, whose values belong to \(\mathscr H_r\). Form the direct sum \(\mathbf H=\mathbf H_1\dotplus \mathbf H_2\). The space \(\mathbf H\) consists of all pairs of the form \(f=\{f(k),f(x)\}\), where \(f(k)\in \mathbf H_1\), \(f(x)\in \mathbf H_2\).
We define the scalar product of vectors in \(\mathbf H\) by the equality
\[ (f,g)=\sum_{k=1}^{\infty} f(k)g^*(k)+\int_{0}^{l}f(x)g^*(x)\,dx,\qquad f,g\in \mathbf H; \]
\(fg^*\) is the scalar product of vectors in \(\mathscr H_r\).
Let \(\{t(k)\}\), \(\{\mathfrak U_k\}\), \(\{p(k)\}\), \(p(x)\), \(\varphi(k)\), \(\varphi(x)\) satisfy the conditions of the preceding section. Define in \(\mathbf H\) the operator \(Tf\) by the equalities
\[ Tf(k)=f(k)t(k)e^{i\varphi(k)} -\sum_{j=k+1}^{\infty} f(j)p(j)\mathfrak U_j\pi^*(j-1)\pi^{*-1}(k)\mathfrak U_k^{*-1}Jp(k)e^{i\varphi(k)} -\int_{0}^{l} f(t)\sqrt{2}\,p(t)\pi^*(t)\pi^{*-1}(k)\mathfrak U_k^{*-1}Jp(k)e^{i\varphi(k)}, \tag{2} \]
\[ Tf(x)=f(x)e^{i\varphi(x)} -2\int_{x}^{l} f(t)p(t)\pi^*(t)\pi^{*-1}(x)Jp(x)e^{i\varphi(x)}\,dt, \tag{3} \]
where
\[ \pi(k)=\sum_{j=1}^{k}\mathfrak U_j t(j)\mathfrak U_j^{-1},\qquad \pi(x)=\pi(k)\big|_{k=\infty}\int_{0}^{x} e^{-p^2(t)J\,dt},\qquad J=\begin{pmatrix} I_p&0\\0&-I_q\end{pmatrix}. \]
Theorem 1. The operator \(T\) has the following properties:
1) \(T\) is a quasiunitary operator of rank \(r\) with signature \((p,q)\), where \(p,q\) are equal, respectively, to the number of \(+1\) and \(-1\) entries of the matrix \(J\).
2) The spectrum of the operator \(T\), not lying on the circle \(|\zeta|=1\), consists of the set of points of the form \(\zeta_k=\rho_k e^{i\varphi(k)}\) \((k=1,2,\ldots)\), where \(\rho_k\) are the eigenvalues of the matrices \(t(k)\), different from unity. The points \(\zeta_k\) are poles of the resolvent.
3) The spectrum of the operator \(T\) lying on the circle \(|\zeta|=1\) coincides with the set \(\mathfrak R\) of values taken by the function \(e^{i\varphi(x)}\) \((0\le x\le l)\). These values may be essentially singular points of the resolvent.
4) We represent the adjoint operator \(T^* f\) in the form
\[ T^* f(k)=f(k)t(k)e^{-i\varphi(k)} -\sum_{j=1}^{k-1} f(j)e^{-i\varphi(j)}p(j)J\mathfrak U_j^{-1}\pi^{-1}(j)\pi(k-1)\mathfrak U_k p(k); \tag{4} \]
\[
T^* f(x)=f(x)e^{-i\varphi(x)}
-2\int_0^x f(t)e^{-i\varphi(t)}p(t)J\varkappa^{-1}(t)\pi(x)p(x)\,dt
\]
\[
-\sum_{j=1}^{\infty} f(j)e^{-i\varphi(j)}p(j)J\mathfrak U_j^{-1}\pi^{-1}(j)\pi(x)\sqrt{2}\,p(x).
\tag{5}
\]
5) The normalized characteristic matrix-function of the operator \(T\) is determined by formula (1).
An operator \(T\) satisfying the conditions of Theorem 1 will be called a triangular model of a quasiunitary operator.
Theorem 2. For every quasiunitary operator \(T\) of rank \(r\) with signature \((p,q)\) satisfying condition 2) of item 1, one can construct a triangular model of the form (2), (3) and a unitary transformation \(U\), possessing the following properties:
1) The operator \(U\) maps \(H\dotminus\mathfrak M_T\) one-to-one onto the space \(H\dotminus\mathfrak M_{\mathbf T}\). (\(\mathfrak M_T,\mathfrak M_{\mathbf T}\) are the supplementary components of \(T\) and \(\mathbf T\), respectively.)
2) The operator \(T\), defined on \(H\dotminus\mathfrak M_T\), is thereby transformed into its model \(\mathbf T=UTU^{-1}\), defined on \(H\dotminus\mathfrak M_{\mathbf T}\).
The theorem follows from the fact that to every quasiunitary operator \(T\) one can associate a normalized characteristic matrix-function \(w_T(\zeta)\). Using the decomposition of \(w_T(\zeta)\) by formula (1), one can construct a triangular model \(\mathbf T\) by formulas (2) and (3). Since, by Theorem 1, \(w_{\mathbf T}(\zeta)\) coincides with \(w_T(\zeta)\), the operators \(T\) and \(\mathbf T\) are unitary-equivalent up to a supplementary component \(\left({}^2\right)\).
We note that Theorem 2 does not follow from the results of \(\left({}^1\right)\), since the Cayley transform of a quasiunitary operator, generally speaking, is not an operator of the class \((i\Omega)\).
- In the present section the question of completeness of the eigenfunctions and associated functions of the operator \(T\) will be solved in the case when \(J=I\). In this case the quasiunitary operator is a contraction operator \(\left({}^5\right)\).
Theorem 3. If \(T\) is a quasiunitary contraction of rank \(r\) with signature \((r,0)\), then the inequality
\[ |\operatorname{Det}\tau|\le \prod_{k=1}^{\infty}|\zeta_k|, \tag{6} \]
holds, where \(\zeta_k\) are the eigenvalues of the operator \(T\); \(\tau\) is the matrix determined by the equality
\[ \tau=\left\|(Te_k,e'_j)\right\|_{k,j=1}^r \]
(\(e_k,e'_j\) are orthonormal bases in \(D_T\) and \(D'_T\), respectively). The system of eigenfunctions and associated functions of the operator \(T\) is complete if and only if equality holds in relation (6).
The proof of this theorem is based on the properties of the triangular model of the operator \(T\).
- As an application of Theorem 2, let us consider the operator \(T\) defined in the Hilbert space \(H\) by the formulas
\[ T e_k=e_{k-1}\quad (k=\pm 1,\pm 2,\ldots);\qquad T e_0=\tau e_{-1},\quad 0<\tau<1, \]
where \(\{e_k\}_{k=-\infty}^{\infty}\) is an orthonormal basis of \(H\).
It is not difficult to verify that \(T\) is a quasi-unitary operator of rank one, and that the characteristic function of the operator \(w_T(\zeta)\) has the form \(w_T(\zeta)\equiv \tau\). The function \(w_T(\zeta)\) can be represented in the form
\[ w_T(\zeta)=\exp\left[-\int_{0}^{\ln \frac{1}{\tau}} \frac{e^{i\varphi(x)}+\zeta}{e^{i\varphi(x)}-\zeta}\,dx\right], \quad \text{where } \varphi(x)=\frac{2\pi x}{\ln \frac{1}{\tau}}. \tag{7} \]
Therefore the triangular model of the operator \(T\) has the form
\[ Tf(x)=f(x)e^{i\varphi(x)}-2\int_{x}^{\ln \frac{1}{\tau}} f(t)e^{x-t}e^{i\varphi(x)}\,dt, \tag{8} \]
where \(\varphi(x)\) is defined by formula (7).
Odessa Pedagogical Institute
named after K. D. Ushinsky
Received
20 II 1956
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