A BASIS OF ROOT VECTORS OF A DISSIPATIVE OPERATOR*

A. S. MARKUS

(Presented by Academician A. N. Kolmogorov, 21 I 1960)

Let (A) be a linear operator acting in a separable Hilbert space (\mathfrak H). A vector (x) is called a root vector of the operator (A), corresponding to the number (\lambda), if there exists a natural number (n) such that ((A-\lambda I)^n x=0). The set (\mathfrak S_\lambda) of all root vectors of the operator (A) corresponding to the number (\lambda) is called the root linear manifold (or, if it is closed, the root subspace) of the operator (A), corresponding to the number (\lambda). The operator (A) is called dissipative if (\operatorname{Im}(Ax,x)\ge 0) for all (x\in\mathfrak D_A).

In the paper of B. R. Mukminov (¹), sufficient conditions were indicated for the system of eigenvectors of a dissipative operator to be a Riesz basis (²) of its closed linear span. I. M. Glazman (³) obtained an elementary proof of this result (not using M. S. Livshits’ triangular model of dissipative operators) under less restrictive conditions. In the present note we indicate certain conditions under which a dissipative operator has a basis of root vectors.

1. In what follows we shall need the notion of a basis of subspaces (⁴). A sequence ({\mathfrak R_k}_1^\infty) of nonzero subspaces is called a basis of the space (\mathfrak H) if every vector (x\in\mathfrak H) is represented uniquely in the form of a series

[
x=\sum_{k=1}^{\infty} x_k,
]

where (x_k\in\mathfrak R_k). If the subspaces (\mathfrak R_k) are pairwise orthogonal, the basis is called orthogonal. If for the sequence of subspaces ({\mathfrak R_k}_1^\infty) there exists a linear bounded continuously invertible operator (A) and an orthogonal basis of subspaces ({\mathfrak M_k}_1^\infty) such that (A\mathfrak M_k=\mathfrak R_k), then, obviously, ({\mathfrak R_k}_1^\infty) is a basis. We shall call it a Riesz basis. The notion of a Riesz basis was introduced and studied for vector bases by N. K. Bari (²). Riesz bases of subspaces were considered by M. K. Fage (⁴) (he called them straightenable). With the aid of the results of the paper of I. M. Gelfand (⁵) it is not difficult to establish that a basis of subspaces is a Riesz basis if and only if every permutation of it is also a basis.

We shall call a sequence of nonzero subspaces ({\mathfrak R_k}_1^\infty) (\omega)-linearly independent if from the equality

[
\sum_{k=1}^{\infty} x_k=0 \quad (x_k\in\mathfrak R_k)
]

it follows that (x_k=0) ((k=1,2,\ldots)). A sequence of nonzero sub-

* The results of the present note were reported by the author at the All-Union Conference on Functional Analysis and Its Applications in Baku in September 1959.

We shall call a sequence of subspaces ({\mathfrak M_k}_1^\infty) separated if, for every natural (k), the minimal angle between the subspace (\mathfrak M_k) and the closed linear span of all the remaining subspaces (\mathfrak M_i) ((i\ne k)) is positive. Recall that the minimal angle (\varphi(\mathfrak M,\mathfrak N)) ((0\le \varphi \le \pi/2)) between subspaces (\mathfrak M) and (\mathfrak N) is defined by the equality ((^6))

[
\cos \varphi(\mathfrak M,\mathfrak N)
=
\sup_{\substack{x\in \mathfrak M,\ y\in \mathfrak N\ |x|=|y|=1}}
|(x,y)|.
]

We note that every separated sequence of subspaces is (\omega)-linearly independent and that every basis is separated.

We shall call two sequences of subspaces ({\mathfrak M_k}_1^\infty) and ({\mathfrak N_k}_1^\infty) quadratically close if

[
\sum_{k=1}^{\infty} |P_k-Q_k|^2 < \infty,
]

where (P_k) ((Q_k)) is the orthogonal projector onto the subspace (\mathfrak M_k) ((\mathfrak N_k)).

The following theorem is an analogue of the well-known theorem of N. K. Bari ((^7)) on the stability of orthonormalized bases.

Theorem 1. A separated sequence of subspaces ({\mathfrak M_k}_1^\infty), quadratically close to some orthogonal basis ({\mathfrak N_k}_1^\infty), is a Riesz basis of its closed linear span.

We note that, unlike the vector bases considered by N. K. Bari, the sequence ({\mathfrak M_k}_1^\infty), generally speaking, is not complete.*

A basis of subspaces quadratically close to some orthogonal basis will be called a Bari basis. The concept of a Bari basis was introduced and studied for vector bases by M. G. Krein ((^8)).

Theorem 2. If a complete separated sequence of subspaces ({\mathfrak M_k}_1^\infty) satisfies the condition

[
\sum_{\substack{j,k=1\ j\ne k}}^{\infty}
\cos^2 \varphi(\mathfrak M_j,\mathfrak M_k) < \infty,
\tag{1}
]

then it is a Bari basis.

This theorem is a partial generalization of the criterion for a Bari basis of vectors obtained by M. G. Krein ((^8)). Unlike vector bases, condition (1) is not necessary, even in the case when the subspaces (\mathfrak M_k) ((k=1,2,\ldots)) are finite-dimensional.**

We note that if the subspaces (\mathfrak M_k) ((k=1,2,\ldots)) appearing in Theorems 1 and 2 are assumed finite-dimensional, then the separability condition may be replaced by the condition of (\omega)-linear independence.

If the sequence of subspaces ({\mathfrak M_k}_1^\infty) is a Riesz basis, then, obviously, the union of orthonormalized bases of the subspaces (\mathfrak M_k) ((k=1,2,\ldots)) is a vector Riesz basis. If, however, the sequence ({\mathfrak M_k}_1^\infty) is a Bari basis, then, as is easy to see, the sequence of vectors composed of orthonormalized bases of the subspaces (\mathfrak M_k) ((k=1,2,\ldots)), while being a Riesz basis, is not, generally speaking, a Bari basis. We shall indicate one sufficient condition under which the indicated sequence of vectors will be a Bari basis.

* If all subspaces (\mathfrak N_k) are finite-dimensional and (\dim \mathfrak N_k=\dim \mathfrak M_k) ((k=1,2,\ldots)), then the sequence ({\mathfrak M_k}_1^\infty) will be complete.

** Condition (1) will be necessary, for example, if the dimensions of all subspaces (\mathfrak M_k) are bounded by one number.

Theorem 3. If ({\mathfrak N_k}_1^\infty) is a complete (\omega)-linearly independent sequence of finite-dimensional subspaces ((n_k=\dim \mathfrak N_k<\infty)), and if

[
\sum_{\substack{j,k=1\ j\ne k}}^\infty
\min(n_j,n_k)\cos^2\varphi(\mathfrak N_j,\mathfrak N_k)<\infty,
]

then the union of the orthonormal bases of the subspaces (\mathfrak N_k) ((k=1,2,\ldots)) is a Bari basis.

1. Let (\lambda) be an eigenvalue of a linear operator (A). By (\nu_\lambda) we shall denote the multiplicity of the eigenvalue (\lambda), i.e. the dimension of the corresponding root subspace (\mathfrak E_\lambda). If there exists a natural number (\mu) such that ((A-\lambda I)^\mu \mathfrak E_\lambda=0), then the least of these numbers will be denoted by (\mu_\lambda); otherwise we put (\mu_\lambda=\infty). The number (\mu_\lambda) will be called the order of the eigenvalue (\lambda). Note that if the operator (A) is closed, then the order (\mu_\lambda) is finite if and only if (\mathfrak E_\lambda) is a subspace.

Lemma. If (A) is a dissipative operator, and (\lambda) and (\tau) are two of its eigenvalues of finite order, then

[
\cos\varphi(\mathfrak E_\lambda,\mathfrak E_\tau)
\le
\frac{2\sqrt{\mu_\lambda\mu_\tau\,\operatorname{Im}\lambda\,\operatorname{Im}\tau}}
{|\lambda-\overline{\tau}|}
\sum_{j=0}^{\mu_\lambda-1} a^j
\sum_{k=0}^{\mu_\tau-1} b^k,
]

where

[
a=\frac{2\sqrt2(\mu_\lambda-1)\operatorname{Im}\lambda}
{|\lambda-\overline{\tau}|},
\qquad
b=\frac{2\sqrt2(\mu_\tau-1)\operatorname{Im}\tau}
{|\lambda-\overline{\tau}|}.
]

This lemma is a generalization of the estimate obtained by I. M. Glazman [3] for eigenvectors.

Theorem 4. Let (A) be a closed linear operator possessing a sequence of eigenvalues ({\lambda_k}_1^\infty) ((\lambda_k\ne\lambda_j) for (k\ne j)). If the following conditions are satisfied:

1) (A=H+iT), where (H) is self-adjoint, and (T) is a nonnegative completely continuous operator;

2) for some natural number (n),

[
\sum_{\substack{j,k=n\ j\ne k}}^\infty
\mu_{\lambda_j}\mu_{\lambda_k}
\frac{\operatorname{Im}\lambda_j\,\operatorname{Im}\lambda_k}
{|\lambda_j-\overline{\lambda_k}|^2}
<\infty;
]

3)

[
\varlimsup_{\substack{j,k\to\infty\ j\ne k}}
\frac{(\mu_{\lambda_j}-1)\operatorname{Im}\lambda_j}
{|\lambda_j-\overline{\lambda_k}|}
<
\frac{1}{2\sqrt2},
]

then the sequence of the corresponding root subspaces ({\mathfrak E_{\lambda_k}}1^\infty) is a Bari basis of its closed linear span (\mathfrak E), and the union of the orthonormal bases of the subspaces (\mathfrak E) ((k=1,2,\ldots)) forms a Riesz basis of the subspace (\mathfrak E).

Let us note that conditions 2) and 3) are satisfied if (T) is an operator of finite trace and (|\lambda_n-\lambda_m|\ge\delta>0) ((n\ne m)). These conditions are also satisfied if the eigenvalues (\lambda_n) and the corresponding orders (\mu_{\lambda_n}) have the following asymptotics:

[
\lambda_n=n^\theta(C+r_n+s_n),
\qquad
\mu_{\lambda_n}=O(n^\delta),
]

where (C) is a real number different from zero; ({r_n}_1^\infty) is a sequence of real numbers converging to zero; (|s_n|=O(n^{-2-\delta})), (\theta\ge 1), and (\delta\ge 0).

We also note that Theorem 4 remains valid if condition 1) is replaced by one of the following conditions:

(1')) (A) is a bounded dissipative operator, and all nonreal eigenvalues (\lambda_n) have finite order and none of them is a limit point of the sequence ({\lambda_n}_1^\infty);

(1'')) (A) is a closed dissipative operator, and all nonreal eigenvalues (\lambda_n) have finite order and are isolated points of the spectrum.

Theorem 5. If conditions 1) and 3) of Theorem 4 are satisfied, while condition 2) is replaced by the following stronger condition:

[
2')\qquad
\sum_{\substack{j,k=n\ j\ne k}}^\infty
\min(\nu_{\lambda_j},\nu_{\lambda_k})\mu_{\lambda_j}\mu_{\lambda_k}
\frac{\operatorname{Im}\lambda_j\,\operatorname{Im}\lambda_k}
{|\lambda_j-\overline{\lambda}_k|^2}<\infty,
]

then the union of the orthonormal bases of the subspaces (\mathfrak{E}_{\lambda_k}) ((k=1,2,\ldots)) is a Bari basis of its closed linear span.

We note that Theorems 4 and 5 remain true if condition 1) is replaced by the following:

(1''')) (A) is a bounded dissipative operator, and all nonreal eigenvalues (\lambda_n) have finite multiplicity.

Let us observe that, although it would be more natural to choose in each root subspace a basis consisting of vectors forming Jordan chains, with such a choice already in the case (\mu_{\lambda_k}\ge 3) ((k=1,2,\ldots)) no conditions imposed on the numbers (\lambda_k,\mu_{\lambda_k},\nu_{\lambda_k}) ((k=1,2,\ldots)) are sufficient to ensure that the sequence obtained by uniting these chains is a basis of its closed linear span. More precisely, whatever the sequences of complex numbers ({a_k}1^\infty) ((\operatorname{Im}a_k>0)) tending to zero and the sequences of natural numbers ({b_k}_1^\infty) and ({c_k}_1^\infty), (c_k\ge b_k\ge 3), one can construct such a completely continuous dissipative operator (A) that for it (\lambda_k=a_k), (\mu) ((k=1,2,\ldots)) which is the union of Jordan chains.}=b_k), (\nu_{\lambda_k}=c_k), and it is impossible to choose a basis of the closed linear span of the subspaces (\mathfrak{E}_{\lambda_k

The author expresses his gratitude to M. G. Krein and I. Ts. Gohberg for their attention to the present work and for a number of valuable critical remarks.

Moldavian Branch
of the Academy of Sciences of the USSR

Received
28 XII 1959

CITED LITERATURE

1. B. R. Mukminov, DAN, 99, No. 4 (1954).
2. N. K. Bari, DAN, 54, No. 5, 383 (1946).
3. I. M. Glazman, UMN, 13, issue 3, 179 (1958).
4. M. K. Fage, DAN, 73, No. 5, 895 (1950); 74, No. 6, 1053 (1950).
5. I. M. Gelfand, Scientific Notes of Moscow State University, issue 148, 4, 224 (1951).
6. M. G. Krein, M. A. Krasnosel’skii, D. P. Mil’man, Collected Papers of the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, No. 11, 97 (1948).
7. N. K. Bari, Mathematical Collection, 14 (56), No. 1–2, 51 (1944).
8. M. G. Krein, UMN, 12, issue 3, 333 (1957).