Citation: Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks[J]. Networks and Heterogeneous Media, 2014, 9(2): 197-216. doi: 10.3934/nhm.2014.9.197
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Let $ \mathcal{H} $ and $ \mathcal{K} $ be complex infinite-dimensional separable Hilbert spaces, and let $ \mathcal{B}(\mathcal{H}, \mathcal{K}) $ (resp., $ \mathcal{C}(\mathcal{H}, \mathcal{K}) $, $ \mathcal{C^{+}}(\mathcal{H}, \mathcal{K}) $) be the set of all bounded (resp., closed, closable) operators from $ \mathcal{H} $ to $ \mathcal{K} $. If $ \mathcal{K} = \mathcal{H} $, we use $ \mathcal{B}(\mathcal{H}) $, $ \mathcal{C}(\mathcal{H}) $, and $ \mathcal{C^{+}}(\mathcal{H}) $ as usual. The domain of $ T\in\mathcal{C}(\mathcal{H}) $ is denoted by $ \mathcal D(T) $. A closed operator $ T $ is said to be left (resp., right) Fredholm if its range $ \mathcal R(T) $ is closed and $ \alpha(T) < \infty $ (resp., $ \beta(T) < \infty $), where $ \alpha(T) $ and $ \beta(T) $ denote the dimension of the null space $ \mathcal N(T) $ and the quotient space $ \mathcal H/\mathcal R(T) $, respectively. $ T $ is said to be Fredholm if it is both left and right Fredholm. If $ T $ is a left or right Fredholm operator, then we define the index of $ T $ by $ ind(T) = \alpha(T)-\beta(T) $. $ T $ is called Weyl if it is a Fredholm operator of index zero. The left essential spectrum, right essential spectrum, essential spectrum, and Weyl spectrum of $ T $ are, respectively, defined by
$ σle(T)={λ∈C:T−λI is not left Fredholm},σre(T)={λ∈C:T−λI is not right Fredholm},σe(T)={λ∈C:T−λI is not Fredholm},σw(T)={λ∈C:T−λI is not Weyl}. $
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Iterates $ T^{2}, T^{3}, \cdots $ of $ T\in\mathcal{C}(\mathcal{H}) $ are defined by $ T^{n}x = T(T^{n-1}x) $ for $ x\in\mathcal D(T^{n}) $ ($ n = 2, 3, \cdots $), where
$ \mathcal D(T^{n}) = \{x:x, Tx, \cdots, T^{n-1}x\in\mathcal D(T)\}. $ |
It is easy to see that
$ \mathcal N(T^{n})\subset\mathcal N(T^{n+1}), \quad \mathcal R(T^{n+1})\subset\mathcal R(T^{n}). $ |
If $ n\geq0 $, we follow the convention that $ T^{0} = I $ (the identity operator on $ \mathcal H $, with $ \mathcal D(I) = \mathcal H $). Then $ \mathcal N(T^{0}) = \{0\} $ and $ \mathcal R(T^{0}) = \mathcal H $. It is also well known that if $ \mathcal N(T^{k}) = \mathcal N(T^{k+1}) $, then $ \mathcal N(T^{n}) = \mathcal N(T^{k}) $ for $ n\geq k $. In this case, the smallest nonnegative integer $ k $ such that $ \mathcal N(T^{k}) = \mathcal N(T^{k+1}) $ is called the ascent of $ T $, and is denoted by $ asc(T) $. If no such $ k $ exists, we define $ asc(T) = \infty $. Similarly, if $ \mathcal R(T^{k}) = \mathcal R(T^{k+1}) $, then $ \mathcal R(T^{n}) = \mathcal R(T^{k}) $ for $ n\geq k $. Thus, we can analogously define the descent of $ T $ denoted by $ des(T) $, and define $ des(T) = \infty $ if $ R(T^{n+1}) $ is always a proper subset of $ R(T^{n}) $. We call $ T $ left (resp., right) Browder if it is left (resp., right) Fredholm and $ asc(T) < \infty $ (resp., $ des(T) < \infty $). The left Browder spectrum, right Browder spectrum, and Browder spectrum of $ T $ are, respectively, defined by
$ σlb(T)={λ∈C:T−λI is not left Browder},σrb(T)={λ∈C:T−λI is not right Browder},σb(T)={λ∈C:T−λI is not Browder}. $
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We write $ \sigma(T), \sigma_{ap}(T) $, and $ \sigma_{\delta}(T) $ for the spectrum, approximate point spectrum, and defect spectrum of $ T $, respectively.
For the spectrum and Browder spectrum, the perturbations of $ 2\times2 $ bounded upper triangular operator matrix $ M_{C} = [AC0B]
$ T_{B} = [AB0D] :\mathcal{D}(A)\oplus \mathcal{D}(D)\subset\mathcal{H}\oplus \mathcal{H}\longrightarrow \mathcal{H} \oplus \mathcal{H} $
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is considered, where $ A\in \mathcal{C}(H) $, $ D\in \mathcal{C}(K) $, and $ B\in \mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) = \{B\in\mathcal{C^{+}}(\mathcal{K}, \mathcal{H}): \mathcal{D}(B)\supset\mathcal{D}(D)\} $. It is obvious that $ T_{B} $ is a closed operator matrix. Our main goal is to present some sufficient and necessary conditions for $ T_{B} $ to be Browder (resp., invertible) for some closable operator $ B\in \mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $ applying a space decomposition technique. Further, the sufficient and necessary condition, which is completely described by the diagonal operators $ A $ and $ D $, for
$ σ∗(TB)=σ∗(A)∪σ∗(D) $
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(1.1) |
is characterized, where $ \sigma_{*} $ is the Browder spectrum (resp., spectrum). This is an extension of the results from [3,7,10,12]. In addition, this result is applied to a Hamiltonian operator matrix from elasticity theory.
Definition 1.1. [4] Let $ T:\mathcal{D}(T)\subset \mathcal{H}\longrightarrow \mathcal{K} $ be a densely defined closed operator. If there is an operator $ T^{\dagger}:\mathcal D(T^{\dagger})\subset \mathcal{K}\longrightarrow \mathcal{H} $ such that $ \mathcal{D}(T^{\dagger}) = \mathcal{R}(T)\oplus\mathcal{R}(T)^{\perp}, \; \mathcal{N}(T^{\dagger}) = \mathcal{R}(T)^{\perp} $ and
$ T^{\dagger}Tx = P_{\overline{\mathcal{R}(T^{\dagger})}}x, x\in\mathcal D(T), \quad TT^{\dagger}y = P_{\overline{\mathcal{R}(T)}}y, y\in\mathcal D(T^{\dagger}) $ |
then $ T^{\dagger} $ is called the maximal Tseng inverse of $ T $.
For the proof of the main results in the next sections, we need the following lemmas:
Lemma 1.1. [2] Let $ T_{B} = [AB0D]
$ (1) $ $ \sigma_{le}(A)\cup\sigma_{re}(D) \subset \sigma_{e}(T_{B}) \subset\sigma_{e}(A)\cup \sigma_{e}(D). $
$ (2) $ $ \sigma_{w}(T_{B}) \subset\sigma_{w}(A)\cup \sigma_{w}(D). $
$ (3) $ $ \sigma_{b}(T_{B}) \subset\sigma_{b}(A)\cup \sigma_{b}(D). $
Lemma 1.2. [17] Let $ T_{B} = [AB0D]
$ (1) $ $ asc(A)\leq asc(T_{B})\leq asc(A)+asc(D), $
$ (2) $ $ des(D)\leq des(T_{B})\leq des(A)+des(D). $
Lemma 1.3. [16] Let $ T\in\mathcal{C}(H) $. Suppose that either $ \alpha(T) $ or $ \beta(T) $ is finite, and that $ asc(T) $ is finite. Then, $ \alpha(T)\leq\beta(T) $.
Lemma 1.4. [6] Let $ T\in\mathcal{C}(H) $. Then, the following inequalities hold for any non negative integer $ k $:
$ (1) $ $ \alpha(T^{k})\leq asc(T_{B})\cdot\alpha(T); $
$ (2) $ $ \beta(T^{k})\leq des(T_{B})\cdot\beta(T) $.
Lemma 1.5. [11] Suppose that the closed operator $ T $ is a Fredholm operator and $ B $ is $ T $-compact. Then,
$ (1) $ $ T+B $ is a Fredholm operator,
$ (2) $ $ ind(T+B) = ind(T) $.
Lemma 1.6. [16] Let $ T\in\mathcal{C}(H) $. Suppose that $ asc(T) $ is finite, and that $ \alpha(T) = \beta(T) < \infty $. Then, $ des(T) = asc(T) $.
In this section, some sufficient and necessary conditions for $ T_{B} $ to be Browder for some closable operator $ B $ with $ \mathcal D(B)\supset \mathcal D(D) $ are given and the set $ \bigcap_{B\in \mathcal{C}^{+}_{D}(\mathcal{K, H})}\sigma_{b}(T_{B}) $ is estimated.
Theorem 2.1. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. If $ \beta(A) = \infty $, then $ T_{B} $ is left Browder for some $ B\in\mathcal{C}_{D}^{+}(\mathcal{K}, \mathcal{H}) $ if and only if $ A $ is left Browder.
Proof. The necessity is obvious by Lemmas 1.1–1.3.
Now we verify the sufficiency. Let $ p = asc(A) < \infty $, then $ \alpha(A^{p})\leq p\cdot\alpha(A) < \infty $ from Lemma 1.4. Thus, we get $ \dim(\mathcal{R}(A)+\mathcal{N}(A^{p}))^{\perp} = \infty $, since $ \beta(A) = \infty $. It follows that there are two infinite dimensional subspaces $ \Delta_{1} $ and $ \Delta_{2} $ of $ (\mathcal{R}(A)+\mathcal{N}(A^{p}))^{\perp} $ such that $ (\mathcal{R}(A)+\mathcal{N}(A^{p}))^{\perp} = \Delta_{1}\oplus\Delta_{2} $. Define an operator $ B:\mathcal{K}\rightarrow \mathcal{H} $ by
$ B = T_{B} = [0U0] : \mathcal{K} \rightarrow [R(A)+N(Ap)Δ1Δ2] $
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where $ U:\mathcal{K}\rightarrow \Delta_{1} $ is a unitary operator. Then, we can claim that $ T_{B} $ is left Browder.
First we prove that $ \mathcal{N}(T_{B}) = \mathcal{N}(A)\oplus \{0\} $, which implies $ \alpha(T_{B}) < \infty $. It is enough to verify that $ \mathcal{N}(T_{B})\subset\mathcal{N}(A)\oplus \{0\} $. Let $ \left(xy
Next, we check that $ \mathcal{R}(T_{B}) $ is closed. We only need to check that the range of $ [U0D]
$ [U0D] y_{n}\rightarrow \left(y10y3 \right)(n \rightarrow \infty) $
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where $ y_{n}\in \mathcal{D}(D) $. Then, $ Uy_{n}\rightarrow y _{1}(n \rightarrow \infty) $ and $ Dy_{n}\rightarrow y_{3}(n \rightarrow \infty) $. Thus, we have that $ y_{n}\rightarrow U^{-1}y_{1}(n\rightarrow \infty) $ from $ U $ is a unitary operator. We also obtain $ U^{-1}y_{1}\in\mathcal{D}(D) $ and $ DU^{-1}y_{1} = y _{3} $ since $ D $ is a closed operator. Hence,
$ \left(y10y3 \right) = [U0D] U^{-1}y_{1} $
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which implies $ \mathcal{R}(T_{B}) $ is closed.
Last, we verify that $ \mathcal{N}(T_{B}^{p}) = \mathcal{N}(T_{B_{0}}^{p+1}) $, which induces $ asc(T_{B}) < \infty $. Let $ \left(xy
$ {Ap+1x+ApBy+Ap−1BDy+⋅⋅⋅+ABDp−1y+BDpy=0,Dp+1y=0. $
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Thus, $ D^{p}y\in \mathcal{N}(D)\cap\mathcal{R}(D^{p}) $ and $ BD^{p}y\in(\mathcal{R}(A)+\mathcal{N}(A^{p}))^{\perp}\subset\mathcal{R}(A)^{\perp} $. This implies that $ A^{p+1}x+A^{p}By+A^{p-1}BDy+\cdot\cdot\cdot+ABD^{p-1}y = -BD^{p}y\in\mathcal{R}(A) \cap(\mathcal{R}(A)+\mathcal{R}(A^{p}))^{\perp}\subset\mathcal{R}(A) \cap\mathcal{R}(A)^{\perp} = \{0\} $, and then
$ {Ap+1x+ApBy+Ap−1BDy+⋅⋅⋅+ABDp−1y=0,BDpy=0. $
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We can obtain $ D^{p}y = 0 $ by the definition of $ B $. According to $ A^{p+1}x+A^{p}By+\cdot\cdot\cdot+ABD^{p-1}y = A(A^{p}x+A^{p-1}By+ \cdot\cdot\cdot+BD^{p-1}y) = 0 $, we have
$ A^{p}x+A^{p-1}By+\cdot\cdot\cdot+BD^{p-1}y\in \mathcal{N}(A). $ |
Let $ x_{1}: = A^{p}x+A^{p-1}By+\cdot\cdot\cdot+BD^{p-1}y $, then
$ A^{p}x+A^{p-1}By+\cdots+ABD^{p-2}y-x_{1}+BD^{p-1}y = 0. $ |
It follows that $ A^{p}x+A^{p-1}By+\cdots+ABD^{p-2}y-x_{1} $= $-BD^{p-1}y\in (\mathcal{R}(A)+\mathcal{N}(A))\cap(\mathcal{R}(A)+$$\mathcal{N}(A^{p}))^{\perp}\subset (\mathcal{R}(A)+\mathcal{N}(A^{p}))\cap(\mathcal{R}(A)+\mathcal{N}(A^{p}))^{\perp} = \{0\} $. Then, $ A^{p}x+A^{p-1}By+\cdots+ABD^{p-2}y-x_{1} = -BD^{p-1}y = 0, $ i.e.,
$ {Apx+Ap−1By+Ap−1BDy+⋅⋅⋅+ABDp−2y=x1,BDp−1y=0. $
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Let $ x_{2}: = A^{p-1}x+A^{p-2}By+A^{p-1}BDy+\cdot\cdot\cdot+BD^{p-2}y $, then $ x_{2}\in\mathcal{N}(A^{2}) $, as $ Ax_{2} = x_{1} $ and $ x_{1}\in\mathcal{N}(A) $. This implies $ A^{p-1}x+A^{p-2}By+A^{p-3}BDy+\cdot\cdot\cdot+BD^{p-3}y-x_{2} =−BDp−2y∈(R(A)+N(A2))
$ {Ap−1x+Ap−2By+Ap−3BDy+⋅⋅⋅+BDp−3y=x2,BDp−2y=0. $
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Continuing this process, there is $ x_{p}\in\mathcal{N}(A^{p}) $ such that
$ {Ax+By=xp,By=0. $
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Thus, $ x\in\mathcal{N}(A^{p+1}) = \mathcal{N}(A^{p}) $. This induces that $ \left(xy
Theorem 2.2. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. If $ \alpha(D) = \infty $, then $ T_{B} $ is right Browder for some $ B\in\mathcal{C}_{D}^{+}(\mathcal{K}, \mathcal{H}) $ if and only if $ D $ is right Browder.
Proof. If $ T_{B} $ is right Browder for some $ B\in\mathcal{C}_{D}^{+}(\mathcal{K}, \mathcal{H}) $, then $ D $ is right Fredholm and $ des(D) < \infty $ by Lemmas 1.1 and 1.2. According to the assumption $ \alpha(D) = \infty $, we can obtain $ ind(D)\geq0 $, which means $ D $ is right Browder.
Now, we verify the reverse implication. Denote $ q = des(D) $. By Lemma 1.4, we have $ \beta(D^{q})\leq q\cdot\beta(D) < \infty $, and then $ \mathcal{R}(D^{q}) $ is closed. Thus, $ \mathcal{K} = \mathcal{R}(D^{q}) \oplus \mathcal{R}(D^{q})^{\perp} $. This includes $ \mathcal{N}(D) = [\mathcal{N}(D)\cap\mathcal{R}(D^{q})]\oplus [\mathcal{N}(D)\cap\mathcal{R}(D^{q})^{\perp}] $. According to the assumption $ \dim(\mathcal{N}(D)) = \infty $ and $ \dim(\mathcal{N}(D)\cap\mathcal{R}(D^{q})^{\perp}) < \infty $, we know that $ \dim(\mathcal{N}(D)\cap\mathcal{R}(D^{q})) = \infty $. Then, there exist two infinite dimensional subspaces $ \Omega_{1} $ and $ \Omega_{2} $ such that $ \mathcal{N}(D)\cap\mathcal{R}(D^{q}) = \Omega_{1}\oplus \Omega_{2} $. Define the operator $ B_{0} $ by
$ B = [U000] : [Ω1Ω2N(D)∩R(Dq)⊥N(D)⊥] \rightarrow \mathcal{H} $
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where $ U:\Omega_{1}\rightarrow \mathcal{H} $ is a unitary operator. Then, we obtain that $ \mathcal{R}(T_{B}) = [HR(D)]
Remark 2.1. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. If $ T_{B} $ is Browder for some $ B\in\mathcal{C}_{D}^{+}(\mathcal{K}, \mathcal{H}) $, then $ \alpha(D) < \infty $ if and only if $ \beta(A) < \infty $. In fact, if $ T_{B} $ is Browder for some $ B\in\mathcal{C}_{D}^{+}(\mathcal{K}, \mathcal{H}) $, then $ A $ is left Fredholm with finite ascent and $ D $ is right Fredholm with finite descent, by Lemmas 1.1 and 1.2. It follows that $ T_{B} $ admits the following representation:
$ T_{B} = [A1B11B120B21B2200D1000] : [D(A)N(D)N(D)⊥∩D(A)] \longrightarrow [R(A)R(A)⊥R(D)R(D)⊥] $
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where $ D_{1}:\mathcal{D}(D)\rightarrow \mathcal{R}(D) $ is invertible and $ A_{1}:\mathcal{D}(A)\rightarrow \mathcal{R}(A_{1})( = \mathcal{R}(A)) $ is an operator with closed range. Let $ A^{\dagger}_{1}:\mathcal{R}(A)\rightarrow \mathcal{H} $ be the maximal Tseng inverse of $ A_{1} $, then $ A_{1}A^{\dagger}_{1} = I_{\mathcal{R}(A)} $. Set $ P = [I0−B12D−1100I−B22D−11000I0000I]
$ PT_{B}Q = [A1000B21000D1000] : = T_{B_{21}}. $
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Thus, $ ind(T_{B}) = ind(T_{B_{21}}) $ since $ P $ and $ Q $ are both injective. Assume that $ \alpha(D) < \infty $ (resp., $ \beta(A) < \infty $), then $ B_{21} $ is a compact operator. By Lemma 1.5, we also obtain $ ind(T_{B}) = ind(A)+ind(D) = 0 $. Therefore, $ \alpha(D) < \infty $ if and only if $ \beta(A) < \infty $.
The next result is given under the hypothesis that $ \beta(A) $ and $ \alpha(D) $ are both finite.
Theorem 2.3. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. Suppose that $ \beta(A) < \infty $ and $ \alpha(D) < \infty $. Then, $ T_{B} $ is Browder for some $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $ if and only if $ A $ is left Browder, $ D $ is right Browder, and $ \alpha(A)+\alpha(D) = \beta(A)+\beta(D) $.
Proof. Necessity. If $ T_{B} $ is Browder for some $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $, then $ A $ is left Fredholm, $ D $ is right Fredholm, $ asc(A) < \infty $, and $ des(D) < \infty $ by Lemmas 1.1 and 1.2. This means $ A $ is left Browder and $ D $ is right Browder. We can also get $ \alpha(A)+\alpha(D) = \beta(A)+\beta(D) $ by Remark 2.1.
Sufficiency. There are two cases to consider.
Case Ⅰ: Assume that $ \alpha(D)\leq\beta(A) $, then we can define a unitary operator $ J_{1}:\mathcal{N}(D)\rightarrow \mathcal{M}_{1} $, where $ \mathcal{M}_{1}\subset\mathcal{R}(A)^{\perp} $ and $ \dim(\mathcal{M}_{1}) = \alpha(D) $. Set
$ B = [00J10] : [N(D)N(D)⊥] \rightarrow [R(A)R(A)⊥] . $
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We have $ \alpha(T_{B}) = \alpha(A) $ and $ \beta(T_{B}) = \beta(D)+\beta(A)-\alpha(D) $. Then, $ asc(T_{B}) = asc(A) < \infty $ and $ \alpha(T_{B}) = \beta(T_{B}) < \infty $ by the assumption $ \alpha(A)+\alpha(D) = \beta(A)+\beta(D) $. Thus, $ des(T_{B}) < \infty $ from Lemma 1.6, which means $ T_{B} $ is Browder.
Case Ⅱ: Suppose that $ \alpha(D) > \beta(A) $. Define a unitary operator $ J_{2}:\mathcal{M}_{2}\rightarrow \mathcal{R}(A)^{\perp} $, where $ \mathcal{M}_{2}\subset\mathcal{N}(D) $ and $ \dim(\mathcal{M}_{2}) = \beta(A) $. Let
$ B = [00J20] : [M2M⊥2] \rightarrow [R(A)R(A)⊥] . $
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We get $ \alpha(T_{B}) = \alpha(A)+\alpha(D)-\beta(A) $ and $ \beta(T_{B}) = \beta(D) $. Then, $ des(T_{B}) = des(D) < \infty $ and $ \alpha(T_{B}) = \beta(T_{B}) < \infty $ by the assumption $ \alpha(A)+\alpha(D) = \beta(A)+\beta(D) $. Hence, $ \alpha(T_{B}^{*}) = \beta(T_{B}^{*}) < \infty $ and $ asc(T_{B}^{*}) = des(T_{B}) < \infty $ by [1, Proposition 3.1]. Therefore, we get $ asc(T_{B}) = asc(T_{B}^{**}) = des(T_{B}^{*})<\infty $ by Lemma 1.6. This means $ T_{B} $ is Browder.
According to Theorems 2.1–2.3 and Remark 2.1, we obtain the theorem below, which is the main result of this section.
Theorem 2.4. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. Then, $ T_{B} $ is Browder for some $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $ if and only if $ A $ is left Browder, $ D $ is right Browder, and $ \alpha(A)+\alpha(D) = \beta(A)+\beta(D) $.
Immediately, we get the following corollary, in which the set $ \bigcap_{B\in \mathcal{C}^{+}_{D}(\mathcal{K, H})}\sigma_{b}(T_{B}) $ is estimated:
Corollary 2.1. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. Then,
$ \bigcap\limits_{B\in \mathcal{C}^{+}_{D}(\mathcal{K, H})}\sigma_{b}(T_{B}) = \sigma_{lb}(A)\cup\sigma_{rb}(D)\cup \{\lambda\in \mathbb{C}:\alpha(A-\lambda I)+\alpha(D-\lambda I) $ |
$ \neq\beta(A-\lambda I)+\beta(D-\lambda I)\} . $ |
Remark 2.2. Theorem 2.4 and Corollary 2.1 extend the results of bounded case in [5,18].
Theorem 2.5. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. Then,
$ σb(TB)=σb(A)∪σb(D) $
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(2.1) |
for every $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $, if and only if $ \lambda\in\sigma_{asc}(D)\setminus(\sigma_{lb}(A)\cup\sigma_{rb}(D)) $ implies $ \alpha(A-\lambda I)+\alpha(D-\lambda I)\neq\beta(A-\lambda I)+\beta(D-\lambda I), $ where $ \sigma_{asc}(D) = \{\lambda\in \mathbb{C}:asc(D-\lambda I) = \infty\} $.
Proof. Equation (2.1) holds for every $ B\in \mathcal{C}^{+}_{D}(\mathcal{K, H}) $ if and only if
$ \sigma_{asc}(D)\setminus(\sigma_{lb}(A)\cup\sigma_{rb}(D))\subset\sigma_{b}(T_{B}) $ |
by [2, Theorem 3.3]. That is,
$ \sigma_{asc}(D)\setminus(\sigma_{lb}(A)\cup\sigma_{rb}(D))\subset \bigcap\limits_{B\in \mathcal{C}^{+}_{D}(\mathcal{K, H})}\sigma_{b}(T_{B}). $ |
This induces
$ \sigma_{asc}(D)\setminus(\sigma_{lb}(A)\cup\sigma_{rb}(D))\subset \{\lambda\in \mathbb{C}:\alpha(A-\lambda I)+\alpha(D-\lambda I)\neq\beta(A-\lambda I)+\beta(D-\lambda I)\}. $ |
Theorem 3.1. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ with dense domains. Then, $ T_{B} $ is right invertible for some $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $ if and only if $ D $ is right invertible and $ \alpha(D)\geq\beta(A) $.
Proof. Necessity. If $ T_{B} $ is right invertible for some $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $, then $ D $ is right invertible and $ T_{B} $ has the following representation:
$ T_{B} = [A1B11B120B21B2200D1] : [D(A)N(D)N(D)⊥∩D(A)] \longrightarrow [R(A)R(A)⊥K] . $
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Obviously, $ D_{1}:\mathcal{N}(D)^{\perp}\cap\mathcal{D}(A)\rightarrow \mathcal{K} $ is a bijection. Set $ P = [I0−B12D−110I−B22D−1100I]
$ PT_{B} = [A1B1100B21000D1] : = \widehat{T}_{B}. $
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Thus, $ \mathcal{R}(\widehat{T}_{B}) = \mathcal{H} \oplus \mathcal{K} $ since $ P $ is bijective, and hence $ \alpha(D)\geq\beta(A) $.
Sufficiency. There are two cases to consider.
Case Ⅰ: Let $ \alpha(D) = \infty $, then we can define a Unitary operator $ U:\mathcal{N}(D)\rightarrow \mathcal{H} $. Set
$ B = [U0] :[N(D)N(D)⊥] \rightarrow \mathcal{H}. $
|
Clearly, $ T_{B} $ is right invertible.
Cases Ⅱ: Assume that $ \alpha(D) < \infty $, then we can define the right invertible operator $ R:\mathcal{N}(D)\rightarrow \mathcal{R}(A)^{\perp} $. Set
$ B = [00R0] :[N(D)N(D)⊥] \rightarrow [R(A)R(A)⊥] . $
|
Clearly, $ T_{B} $ is right invertible.
From the above theorem and Theorems 5.2.1, 5.2.3 of [15], we can obtain the next results immediately.
Theorem 3.2. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ with dense domains. Then, $ T_{B} $ is invertible for some $ B\in\mathcal{C}^{+}_{D}(\mathcal{K}, \mathcal{H}) $ if and only if $ A $ is left invertible, $ D $ is right invertible, and $ \alpha(D)\geq\beta(A) $.
Corollary 3.1. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ with dense domains. Then,
$ \bigcap\limits_{B\in \mathcal{C}^{+}_{D}(\mathcal{K, H})}\sigma(T_{B}) = \sigma_{ap}(A)\cup\sigma_{\delta}(D)\cup \{\lambda\in \mathbb{C}:\alpha(D-\lambda I)\neq\beta(A-\lambda I)\}. $ |
Remark 3.1. Theorem 3.2 and Corollary 3.1 are also valid for bounded operator matrix $ B\in\mathcal{B}(\mathcal{K}, \mathcal{H}) $. These conclusions extend the results in [8,9,12,13].
From Corollary 3.1, we have the following theorem, which extends the results of [3,7,10,12].
Theorem 3.3. Let $ A\in\mathcal{C}(\mathcal{H}) $ and $ D\in\mathcal{C}(\mathcal{K}) $ be given operators with dense domains. Then,
$ σ(TB)=σ(A)∪σ(D) $
|
(3.1) |
for every $ B\in \mathcal{C}^{+}_{D}(\mathcal{K, H}) $ if and only if
$ \lambda\in \sigma_{p, 1}(D)\cap \sigma_{r, 1}(A) \Rightarrow \alpha(D-\lambda I)\neq\beta(A-\lambda I) $ |
where $ \sigma_{p, 1}(D) = \{\lambda\in\mathbb{C}:\mathcal{N}(D-\lambda I)\neq\{0\}, \mathcal{N}(D-\lambda I) = \mathcal{K}\} $ and $ \sigma_{r, 1}(A) = \{\lambda\in\mathbb{C}:\mathcal{N}(A-\lambda I) = \{0\}$, $\mathcal{R}(A-\lambda I) = \overline{\mathcal{R}(A-\lambda I)}\neq\mathcal{H}\} $.
Proof. Equation (3.1) holds for every $ B\in \mathcal{C}^{+}_{D}(\mathcal{K, H}) $ if and only if
$ \sigma_{p, 1}(D)\cap \sigma_{r, 1}(A)\subset\sigma(T_{B}) $ |
by [7, Corollary 2]. That is,
$ \sigma_{p, 1}(D)\cap \sigma_{r, 1}(A)\subset \bigcap\limits_{B\in \mathcal{C}^{+}_{D}(\mathcal{K, H})}\sigma(T_{B}). $ |
This induces
$ \sigma_{p, 1}(D)\cap \sigma_{r, 1}(A)\subset \{\lambda\in \mathbb{C}:\alpha(D-\lambda I)\neq\beta(A-\lambda I)\} $ |
by Corollary 3.1, $ \sigma_{r, 1}(A)\cap\sigma_{ap}(A) = \emptyset $, and $ \sigma_{p, 1}(D)\cap\sigma_{\delta}(D) = \emptyset $.
Applying Theorems 2.5 and 3.3 for the Hamiltonian operator matrix, we obtain the next result.
Theorem 3.4. Let $ H = [AB0−A∗]
$ (1) $ For every $ C\in \mathcal{C}^{+}_{A^{*}}(\mathcal{K, H}) $,
$ σb(H)=−σb(A)∗∪σb(A) $
|
(3.2) |
if and only if
$ \lambda\in\sigma_{asc}(A^{*})\setminus(-\sigma_{lb}(A)\cup\sigma_{rb}(A^{*}))\Rightarrow \alpha(A-\lambda I)+\alpha(A+\overline{\lambda} I)\neq\beta(A-\lambda I)+\beta(A+\overline{\lambda} I). $ |
$ (2) $ For every $ C\in \mathcal{C}^{+}_{A^{*}}(\mathcal{K, H}) $,
$ σ(H)=−σ(A)∗∪σ(A) $
|
(3.3) |
if and only if
$ \lambda\in \left(-\sigma_{r, 1}(A)^{*}\cap \sigma_{r, 1}(A)\right)\setminus\{\lambda\in\mathbb{C}:Re\lambda = 0\}\Rightarrow \beta(A+\overline{\lambda} I)\neq\beta(A-\lambda I). $ |
In particular, if $ \sigma_{r, 1}(A) $ does not include symmetric points about the imaginary axis, then (3.3) holds.
Example 3.1. Consider the plate bending equation in the domain $ \{(x, y): 0 < x < 1, 0 < y < 1\} $
$ D(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}})^{2}\omega = 0 $ |
with boundary conditions
$ \omega(x, 0) = \omega(x, 1) = 0, $ |
$ \frac{\partial^{2}\omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}} = 0, y = 0, 1. $ |
Set
$ \theta = \frac{\partial\omega}{\partial x}, q = D(\frac{\partial^{3}\omega}{\partial x^{3}}+\frac{\partial^{3}\omega}{\partial y^{3}}), m = -D(\frac{\partial^{2}\omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}}) $ |
then the equation can be written as the Hamiltonian system (see [19])
$ \frac{\partial}{\partial x}\left(ωθqm \right) = [0100−∂2∂y200−1D000∂2∂y200−10] \left(ωθqm \right) $
|
and the corresponding Hamiltonian operator matrix is given by
$ H = [0100−d2dy200−1D000d2dy200−10] = [AB0−A∗] $
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with domain $ \mathcal{D}(A)\oplus \mathcal{D}(A^{*})\subset \mathcal{H}\oplus \mathcal{H} $, where $ \mathcal H = L_{2}(0, 1)\oplus L_{2}(0, 1) $, $ \mathcal A = AC[0, 1] $, and
$ A = [01−d2dy20] , B = [000−1D] , $
|
$ \mathcal D(A) = \left\{\left(ωθ \right) \in \mathcal H:\omega(0) = \omega(1) = 0, \omega^{'}\in \mathcal A, \omega^{''}\in \mathcal H\right\}. $
|
By a simple calculation, we get $ \sigma_{r, 1}(A) = \emptyset $ and $ \sigma(A) = \{k\pi:k = \pm1, \pm2, ...\}. $ By Theorem 3.4, we get
$ \sigma(H) = -\sigma(A^{*})\cup\sigma(A) = \{k\pi:k = \pm1, \pm2, ...\}. $ |
On the other hand, we can easily calculate that
$ \sigma(H) = \{k\pi:k = \pm1, \pm2, ...\} = -\sigma(A^{*})\cup\sigma(A). $ |
Certain spectral and Browder spectral properties of closed operator matrix $ T_{B} = [AB0D]
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
We are grateful to the referees for their valuable comments on this paper.
This work is supported by the Basic Scientific Research Funds of Subordinate Universities of Inner Mongolia (Grant No. ZSLJ202213).
No potential conflict of interest was reported by the authors.
[1] |
W. J. Anderson, Continuous-Time Markov Chains. An Application Oriented Approach, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0
![]() |
[2] |
W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349. doi: 10.1112/plms/s3-54.2.321
![]() |
[3] | J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications, Springer Verlag, London, 2006. |
[4] |
J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001
![]() |
[5] |
J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1
![]() |
[6] | N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall, Inc., Englewood Cliffs, 1974. |
[7] | B. Dorn, Flows in Infinite Networks - A Semigroup Aproach, Ph.D thesis, University of Tübingen, 2008. |
[8] |
B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2
![]() |
[9] |
B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012
![]() |
[10] |
B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x
![]() |
[11] | K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 2000. |
[12] | F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, Inc. New York, 1959. |
[13] |
Ch. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York, 2001. doi: 10.1007/978-1-4613-0163-9
![]() |
[14] |
F.M. Hante, G. Leugering and T. I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. & Optimization, 59 (2009), 275-292. doi: 10.1007/s00245-008-9057-6
![]() |
[15] |
M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3
![]() |
[16] |
T. Matrai and E. Sikolya, Asymptotic behaviour of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018
![]() |
[17] |
C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512
![]() |
[18] | H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. |
[19] | R. Nagel, ed., One-parameter Semigroups of Positive Operators, Springer Verlag, Berlin, 1986. |
[20] | P. Namayanja, Transport on Network Structures, Ph.D thesis, UKZN, 2012. |
[21] |
E. Seneta, Nonnegative Matrices and Markov Chains, Springer Verlag, New York, 1981. doi: 10.1007/0-387-32792-4
![]() |
[22] | E. Sikolya, Semigroups for Flows in Networks, Ph.D dissertation, University of Tübingen, 2004. |
[23] |
E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z
![]() |