����������� ������ 54 (2002), 83–88

UDK 517.984.3

���������� ������ ��� research paper

ON DAVIS-KAHAN-WEINBERGER EXTENSION THEOREM

Dragan S. -Dor - dević

Abstract. If R = [ \begin{matrix}H \\ B \end{matrix} ], where H = H^{∗}, we find a pseudo-inverse form of all solutions W = W^{∗}, such that ∥A∥ = ∥R∥, where A = [ \begin{matrix}H & B^{∗} \\ B & W \end{matrix} ] and ∥H∥ ≤ ∥R∥. In this paper we extend well-known results in a finite dimensional setting, proved by Dao-Sheng Zheng [15]. Thus, a pseudo inverse form of solutions of the Davis-Kahan-Weinberger theorem is established.

1. Motivation

Let Z denote an arbitrary Hilbert space and let H and K denote closed mutually orthogonal subspaces of Z, such that Z = H ⊕ K. We use L(H, K) to denote the set of all bounded operators from H into K and L(H) = L(H, H). For T ∈ L(H, K) let R(T) and N (T), respectively, denote the range and the kernel of T.

Let H = H^{∗} ∈ L(H) and B ∈ L(H, K) be given operators, such that ρ = ∥R∥, where

R = [ \begin{matrix}H \\ B \end{matrix} ] : [ H] → [ \begin{matrix}H \\ K \end{matrix} ] .

Notice that ∥H∥ ≤ ∥R∥ always holds. We consider the following problem. Find an operator W = W^{∗} ∈ L(K), such that the selfadjoint operator

A = [ \begin{matrix}H & B^{∗} \\ B & W \end{matrix} ] : [ \begin{matrix}H \\ K \end{matrix} ] → [ \begin{matrix}H \\ K \end{matrix} ]

satisfies the norm condition ∥A∥ = ∥R∥ = ρ.

This is a typical selfadjoint dilation problem. We mention that a nonselfadjoint form is also important.

The result which is known as the Davis-Kahan-Weinberger theorem is proved in [5, Theorem 1.2] and stated as follows:

AMS Subject Classification: 47 A 05, 47 A 20, 15 A 09

Keywords and phrases: Davis-Kahan-Weinberger theorem, Moore-Penrose inverse.

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

This work was supported by the Ministry of Science, Technology and Development of Serbia, grant no. 1232.

Theorem (DKW). Let H, B, C satisfy ∥[ \begin{matrix}H \\ B \end{matrix} ]∥ ≤ μ, ∥[^{H} ^{C}]∥ ≤ μ and ∥H∥ <

μ. Then there exists W such that ∥[ \begin{matrix}H & C \\ B & W \end{matrix} ]∥ ≤ μ. Indeed those W which have this property are exactly those of the form

W = −KH^{∗}L + μ(I − KK^{∗})^{1/2}Z(I − L^{∗}L)^{1/2},

where K^{∗} = (μ^{2}I − H^{∗}H)^{−1/2}B^{∗}, L = (μ^{2} − HH^{∗})^{−1/2}C and Z is an arbitrary contraction. If H is compact then W may be chosen compact.

The selfadjoint version of the previous theorem follows (see [5, Corollary 1.3]):

Corollary (DKW-SA). Let H be selfadjoint and ∥[ \begin{matrix}H \\ B \end{matrix} ]∥ ≤ μ and ∥H∥ < μ.

Then there exists selfadjoint W such that ∥[ \begin{matrix}H & B^{∗} \\ B & W \end{matrix} ]∥ ≤ μ. Indeed those W which have this property are exactly those such that

−μI + B(μI + H)^{−1}B^{∗} ≤ W ≤ μI − B(μI − H)^{−1}B^{∗}.

The following result is a central solution obtained from Corollary (DKW-SA) (see [5, (1.7)]). One strightforward proof of this result is given in [15, Lemma 3.1] (although the proof is given for complex matrices, a careful reading shows that it is valid for operators on arbitrary Hilbert spaces also).

Corollary (DKW-central). Let R = [ \begin{matrix}H \\ B \end{matrix} ] : [H] → [ \begin{matrix}H \\ K \end{matrix} ], where H = H^{∗},

σ ≥ ∥R∥ and σ > ∥H∥. If W_{σ} = −BH(σ^{2} − H^{2})^{−1}B^{∗} and

A_{σ} = [ \begin{matrix}H & B^{∗} \\ B & W_{σ} \end{matrix} ] ,

then ∥A_{σ}∥ ≤ σ.

A selfadjoint part of this problem is proved by M. G. Krein (see [9] and [13, Sec. 125]). One special case of the Davis-Kahan-Weinberger theorem was proved by B. Sz.-Nagy and C. Foias (see [14, Theorem 1] and also [3]). Several proofs of Theorem (DKW) are presented in [4, Sec. 3], [5, Theorem 1.2] and [12, Theorem 1].

The boundary case appears if we assume ∥H∥ = ∥R∥ = μ. One solution (as a non-selfadjoint extension) is found in [4, Sec. 3]. In this case at least one of μI − H and μI + H is not invertible, but we can consider their Moore-Penrose inverses (in the case when they exist). Zheng used this idea in [15, Theorem 4.1] and completely solved this problem in finite dimensional settings. Kahan also found one solution of this problem, but he did not publish his results, which appeared in [11, p. 231–233] without any proof. See also results of Fioas and Frazho [6, Chapter IV]. Zhang also proved Theorem (DKW-central) in finite dimensional settings, under the more general assumption ∥H∥ ≤ μ. Finally, we mention that finite-dimensional dilation results of this type have lots of applications in numerical analysis (see [5], [7], [8] and [10]).

In this paper we extend Zheng’s results for operators on arbitrary Hilbert spaces.

2. Notations

We use notations in the same way as in [15].

Recall that an operator T^{†} ∈ L(K, H) is the Moore-Penrose inverse of T ∈ L(H, K), if the following is satisfied:

T T^{†}T = T, T^{†}T T^{†} = T^{†}, (T T^{†})^{∗} = T T^{†}, (T^{†}T)^{∗} = T^{†}T.

It is well-known that T^{†} exists if and only if R(T) is closed, and in this case T^{†} is unique [2].

Assume that T ∈ L(H) and 0 is not the point of accumulation of the spectrum σ(T) of T. If the point {0} is the pole of the resolvent λ ↦ (λ − T)^{−1}, then the order of this pole is the Drazin index (or the index) of T, denoted by ind(T). Notice that ind(T) < ∞ holds if and only if there exists the Drazin inverse of T, i.e. there exists the unique operator T^{D} ∈ L(H), such that the following hold:

T^{D}T T^{D} = T^{D}, T T^{D} = T^{D}T, T^{n+1}T^{D} = T^{n}

and the least n in the previous definition is equal to ind(T). If ind(T) ≤ 1, then T^{D} is known as the group inverse of T, denoted by T^{#}. If ind(T) = 0, then T is invertible and T^{−1} = T^{D}.

In this article the group inverse is of special interest. If ind(T) ≤ 1, then H = R(T) +^{•} N (T) and this sum is not necessarily orthogonal. Also, T has the matrix form with respect to this decomposition:

T = [ \begin{matrix}0 & 0 \\ 0 & T_{1} \end{matrix} ] : [ \begin{matrix}N (T) \\ R(T) \end{matrix} ] → [ \begin{matrix}N (T) \\ R(T) \end{matrix} ] ,

where T_{1} = T |_{R(T)} : R(T) → R(T) is invertible [2].

In the case when T is selfadjoint and has a closed range, the Moore-Penrose inverse coincides with the group inverse of T. Also, R(T) is closed if and only if 0 is not the accumulation point of σ(T). In this case the decomposition H = R(T) ⊕ N (T) is orthogonal.

If T = T^{∗} ∈ L(H), then we write T ≥ 0 if and only if (T x, x) ≥ 0 for all x ∈ H, where (·, ·) is the inner product in H. Also, T > 0 if and only if T ≥ 0 and T is invertible.

3. Results

The following result is proved in [1].

Lemma 3.1. Let

S = [ \begin{matrix}S_{11} & S_{12} \\ S_{12}^{∗} & S_{22} \end{matrix} ] : [ \begin{matrix}H \\ K \end{matrix} ] → [ \begin{matrix}H \\ K \end{matrix} ] ,

where S_{11} = S_{11}^{∗}, S_{22} = S_{22}^{∗} and R(S_{11}) is closed. Then S ≥ 0 if and only if the following is satisfied:

S_{11} ≥ 0, S_{11}S_{11}^{†}S_{12} = S_{12} and S_{22} − S_{12}^{∗}S_{11}^{†}S_{12} ≥ 0. Although the original proof in [1] is given for finite dimensional spaces H and

K, the result is valid in infinite dimensional settings also.

We now prove the first auxiliary result.

Lemma 3.2. Let R = [ \begin{matrix}H \\ B \end{matrix} ] : [ H] → [ \begin{matrix}H \\ K \end{matrix} ], H = H^{∗} and ρ = ∥R∥. Then N (ρ−H) ⊂ N (B), R(ρ−H) ⊃ R(B^{∗}), N (ρ+H) ⊂ N (B) and R(ρ+H) ⊃ R(B^{∗}).

Proof. Obviously, ∥H∥ ≤ ρ. Let x ∈ N (ρ − H) and ∥x∥ = 1. Then
ρ^{2} ≥ ∥Rx∥^{2} = ∥Hx∥^{2} + ∥Bx∥^{2} = ρ^{2} + ∥Bx∥^{2},

implying Bx = 0. The rest of the proof is similar. Notice that if there exists any x ∈ N (ρ − H) and ∥x∥ = 1, then ∥H∥ = ρ = ∥R∥.

The following result represents a pseudo inverse form of solutions of the Davis-Kahan-Weinberger theorem.

Theorem 3.3. Let R = [ \begin{matrix}H \\ K \end{matrix} ] : [H] → [ \begin{matrix}H \\ K \end{matrix} ], H = H^{∗}, ρ = ∥R∥, W = W^{∗} ∈ L(K), A = [ \begin{matrix}H & B^{∗} \\ B & W \end{matrix} ] and let R(ρ − H) and R(ρ + H) be closed. Then ∥A∥ = ρ if and only if

B(ρ + H)^{†}B^{∗} − ρ ≤ W ≤ ρ − B(ρ − H)^{†}B^{∗}.

Proof. Obviously, ρ = ∥R∥ ≤ ∥A∥. Since A = A^{∗}, in order to prove ∥A∥ ≤ ρ, it is enough to prove ρ − A ≥ 0 and ρ + A ≥ 0. Notice that

ρ − A = [ \begin{matrix}ρ − H & −B^{∗} \\ −B & ρ − W \end{matrix} ] .

From Lemma 3.1 we know that ρ − A ≥ 0 if and only if:

(1) ρ − H ≥ 0;

(2) (ρ − H)(ρ − H)^{†}B^{∗} = B^{∗};

(3) ρ − W − (−B)(ρ − H)^{†}(−B^{∗}) ≥ 0. We know that (1) always holds. The condition (2) is equivalent to R(B^{∗})

⊂ R(ρ−H), which is always true according to Lemma 3.2. Finally, (3) is equivalent to ρ − B(ρ − H)^{†}B^{∗} ≥ W.

Similarly, ρ + A ≥ 0 is equivalent to B(ρ + H)^{†}B^{∗} − ρ ≤ W. Now we prove the extension of Corollary (DKW-central).

Theorem 3.4. Let R = [ \begin{matrix}H \\ B \end{matrix} ], H = H^{∗}, ρ = ∥R∥ and let R(ρ − H) and R(ρ + H) be closed. If

W = −BH(ρ^{2} − H^{2})^{†}B^{∗} and A = [ \begin{matrix}H & B^{∗} \\ B & W \end{matrix} ] ,

then ∥A∥ = ρ = ∥R∥.

Proof. The case ρ = 0 is trivial. Hence, assume ρ > 0. Since the Moore-Penrose inverse of a selfadjoint operator coincides with its group inverse, we conclude that the decomposition H = N (ρ − H) ⊕ R(ρ − H) is orthogonal and

ρ − H = [ \begin{matrix}0 & 0 \\ 0 & M \end{matrix} ] : [ \begin{matrix}N (ρ − H) \\ R(ρ − H) \end{matrix} ] → [ \begin{matrix}N (ρ − H) \\ R(ρ − H) \end{matrix} ] ,

where M is invertible and M > 0. We conclude that H and ρ+H have the following matrix forms with respect to the same decomposition of H:

H = [ \begin{matrix}ρ & 0 \\ 0 & ρ − M \end{matrix} ] , ρ + H = [ \begin{matrix}2ρ & 0 \\ 0 & 2ρ − M \end{matrix} ] .

Since ρ + H ≥ 0, we conclude 0 < M ≤ 2ρ. From ind(ρ + H) ≤ 1 we conclude that ind(2ρ − M) ≤ 1. Now, R(ρ − H) = N (2ρ − M) ⊕ R(2ρ − M) and this decomposition is orthogonal, since 2ρ − M is selfadjoint. Also

2ρ − M = [ \begin{matrix}0 & 0 \\ 0 & N \end{matrix} ] : [ \begin{matrix}N (2ρ − M) \\ R(2ρ − M) \end{matrix} ] → [ \begin{matrix}N (2ρ − M) \\ R(2ρ − M) \end{matrix} ] ,

where N is invertible. Since M ≤ 2ρ we conclude N > 0. Notice that M =

[ \begin{matrix}2ρ & 0 \\ 0 & 2ρ − N \end{matrix} ], hence from M > 0 we get 0 < N < 2ρ. Finally, we get

H = [ \begin{matrix}ρ & 0 & 0 \\ 0 & −ρ & 0 \\ 0 & 0 & N − ρ \end{matrix} ] , ρ − H = [ \begin{matrix}0 & 0 & 0 \\ 0 & 2ρ & 0 \\ 0 & 0 & 2ρ − N \end{matrix} ] , ρ + H = [ \begin{matrix}2ρ & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & N \end{matrix} ]

and conclude N (ρ+H) = N (2ρ−M). From Lemma 3.2 we know that N (ρ−H) ⊂ N (B) and N (ρ + H) ⊂ N (B), implying the following decomposition of B:

B = [ 0 0 B_{1} ] : [ \begin{matrix}N (ρ − H) \\ N (ρ + H) \\ R(2ρ − M) \end{matrix} ] → K

and also the matrix form of R:

R = [ \begin{matrix}ρ & 0 & 0 \\ 0 & −ρ & 0 \\ 0 & 0 & H_{1} \\ 0 & 0 & B_{1} \end{matrix} ] : [ \begin{matrix}N (ρ − H) \\ N (ρ + H) \\ R(2ρ − M) \end{matrix} ] → [ \begin{matrix}N (ρ − H) \\ N (ρ + H) \\ R(2ρ − M) \\ K \end{matrix} ] ,

where H_{1} = N − ρ. Notice that −ρ < H_{1} < ρ. If P_{R(2ρ−M)} is the orthogonal projection from H onto R(2ρ − M), and P_{R(2ρ−M)⊕K} is the orthogonal projection from H onto R(2ρ − M) ⊕ K, then

R_{1} = [ \begin{matrix}H_{1} \\ B_{1} \end{matrix} ] = P_{R(2ρ−M)⊕K}RP_{R(2ρ−M)},

implying ∥R_{1}∥ ≤ ∥R∥.

Let W_{ρ} = −B_{1}H_{1}(ρ^{2} −H_{1}^{2})^{−1}B_{1}^{∗} and A_{ρ} = [ \begin{matrix}H_{1} & B_{1}^{∗} \\ B_{1} & W_{ρ} \end{matrix} ]. From Lemma (DKW-

central) we know that ∥A_{ρ}∥ ≤ ∥R_{1}∥ ≤ ∥R∥ = ρ.

Now we have the matrix form of A:

A = [ \begin{matrix}ρ & 0 & 0 & 0 \\ 0 & −ρ & 0 & 0 \\ 0 & 0 & H_{1} & B_{1}^{∗} \\ 0 & 0 & B_{1} & W_{ρ} \end{matrix} ] = [ \begin{matrix}ρ & 0 & 0 \\ 0 & −ρ & 0 \\ 0 & 0 & A_{ρ} \end{matrix} ] .

It is easy to see that ∥A∥ = ρ.

We only have to prove the equality BH(ρ^{2} − H^{2})^{†}B^{∗} = B_{1}H_{1}(ρ^{2} − H_{1}^{2})^{−1}B_{1}^{∗}. Since ρ and −ρ are not accumulation points of the spectrum σ(H), we conclude that ρ^{2} is not the accumulation point of H^{2}. Hence, (ρ^{2} − H^{2})^{†} exists. Now we compute

BH(ρ^{2} − H^{2})^{†}B^{∗} =

= [ 0 0 B_{1} ] [ \begin{matrix}ρ & 0 & 0 \\ 0 & −ρ & 0 \\ 0 & 0 & H_{1} \end{matrix} ] [ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & N^{−1}(2ρ − N)^{−1} \end{matrix} ] [ \begin{matrix}0 \\ 0 \\ B_{1}^{∗} \end{matrix} ]

= B_{1}H_{1}N^{−1}(2ρ − N)^{−1}B_{1}^{∗} = B_{1}H_{1}(ρ^{2} − H_{1}^{2})^{−1}B_{1}^{∗}. As a corollary, we get the following result, which cannot be verified easily by

a direct computation.

Corollary 3.5. If R(ρ − H) and R(ρ + H) are closed, where ρ = ∥R∥,

R = [ \begin{matrix}H \\ B \end{matrix} ] and H = H^{∗}, then

B(ρ + H)^{†}B^{∗} − ρ ≤ −BH(ρ^{2} − H^{2})^{†} ≤ ρ − B(ρ − H)^{†}B^{∗}.

Thus, we extended Zheng’s results in [15, Theorem 4.1 and Theorem 4.2].

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(received 30.12.2002)

Department of Mathematics, Faculty of Sciences, University of Niš, P.O. Box 224, 18000 Niš, Serbia, E-mail: dragan@pmf.ni.ac.yu, ganedj@EUnet.yu