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Preserving of the unconditional basis property under non-self-adjoint perturbations of self-adjoint operators
Spectral Theory · Submitted 29 Jan 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Main result of the work consists in proving that the system of the invariant subspaces $Q_k(H)$, $|k|\geq N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.
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Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$ \alpha_{2k+1}-\alpha_{2k} \geq b|\alpha_{2k+1}+\alpha_{2k}|^p\quad \text{ for some }\,b>0,\,p\in[0,1). $$ Suppose that a linear operator $B$ in $H$ is $p$-subordinated to $T$, i.e. $\mathcal D(B) \supset\mathcal D(T)$ and $\|Bx\| \leq b'\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal D(T)$, with some $b'>0$ and $M\geq 0$. Then the spectrum of the perturbed operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double parabola $P_{p,h} = \bigl\{\lambda \in \mathbb{C}\,\bigl|\,|\mathop{\rm Im} \lambda|\leq h|\mathop{\rm Re} \lambda|^p\bigr\}$, provided that $h>b'$. The vertical strips $\Omega_k =\{\lambda\in\mathbb{C}|\,|r_k-{\rm Re}\,\lambda|\leq \delta r_k^p\}$, $r_k =(\alpha_{2k}+\alpha_{2k+1})/2$, belong to the resolvent set of $T$, provided that $\delta <b -b'$ and $ |k|\geq N$ for $N$ large enough. For $|k|\ge N+1$, denote by $\Pi_k$ the curvilinear trapezoid formed by the lines ${\rm Re}\,\lambda = r_{k-1}$, ${\rm Re}\,\lambda = r_{k}$, and the boundary of the parabola $P_{p,h}$. Assume that $Q_0$ is the Riesz projection corresponding to the (bounded) part of the spectrum of $T$ that lies outside $\bigcup_{|k|\geq N+1}\Pi_k$. And let $Q_k$, $|k|\ge N+1$, be the Riesz projection for the part of the spectrum of $T$ confined within $\Pi_k$. Main result of the work consists in proving that the system of the invariant subspaces $Q_k(H)$, $|k|\geq N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. | background 1.00 | 0.08 | 0.01 | asserted 0.90 |
| s1 | Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$ \alpha_{2k+1}-\alpha_{2k} \geq b|\alpha_{2k+1}+\alpha_{2k}|^p\quad \text{ for some }\,b>0,\,p\in[0,1). | background 0.94 | 0.17 | 0.01 | asserted 0.87 |
| s2 | $$ Suppose that a linear operator $B$ in $H$ is $p$-subordinated to $T$, i.e. $\mathcal D(B) \supset\mathcal D(T)$ and $\|Bx\| \leq b'\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal D(T)$, with some $b'>0$ and $M\geq 0$. | background 0.84 | 0.18 | 0.01 | asserted 0.90 |
| s3 | Then the spectrum of the perturbed operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double parabola $P_{p,h} = \bigl\{\lambda \in \mathbb{C}\,\bigl|\,|\mathop{\rm Im} \lambda|\leq h|\mathop{\rm Re} \lambda|^p\bigr\}$, provided that $h>b'$. | result 1.00 | 0.84 | 0.05 | asserted 1.00 |
| s4 | The vertical strips $\Omega_k =\{\lambda\in\mathbb{C}|\,|r_k-{\rm Re}\,\lambda|\leq \delta r_k^p\}$, $r_k =(\alpha_{2k}+\alpha_{2k+1})/2$, belong to the resolvent set of $T$, provided that $\delta <b -b'$ and $ |k|\geq N$ for $N$ large enough. | result 0.78 | 0.68 | 0.01 | asserted 0.98 |
| s5 | For $|k|\ge N+1$, denote by $\Pi_k$ the curvilinear trapezoid formed by the lines ${\rm Re}\,\lambda = r_{k-1}$, ${\rm Re}\,\lambda = r_{k}$, and the boundary of the parabola $P_{p,h}$. | method 0.84 | 0.57 | 0.01 | asserted 0.80 |
| s6 | Assume that $Q_0$ is the Riesz projection corresponding to the (bounded) part of the spectrum of $T$ that lies outside $\bigcup_{|k|\geq N+1}\Pi_k$. | method 0.83 | 0.39 | 0.02 | asserted 0.90 |
| s7 | And let $Q_k$, $|k|\ge N+1$, be the Riesz projection for the part of the spectrum of $T$ confined within $\Pi_k$. | method 0.74 | 0.50 | 0.02 | asserted 0.89 |
| s8 | Main result of the work consists in proving that the system of the invariant subspaces $Q_k(H)$, $|k|\geq N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$. | result 0.83 | 0.94 | 0.98 | asserted 1.00 |
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Original abstract
Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$ \alpha_{2k+1}-\alpha_{2k} \geq b|\alpha_{2k+1}+\alpha_{2k}|^p\quad \text{ for some }\,b>0,\,p\in[0,1). $$ Suppose that a linear operator $B$ in $H$ is $p$-subordinated to $T$, i.e. $\mathcal D(B) \supset\mathcal D(T)$ and $\|Bx\| \leq b'\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal D(T)$, with some $b'>0$ and $M\geq 0$. Then the spectrum of the perturbed operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double parabola $P_{p,h} = \bigl\{\lambda \in \mathbb{C}\,\bigl|\,|\mathop{\rm Im} \lambda|\leq h|\mathop{\rm Re} \lambda|^p\bigr\}$, provided that $h>b'$. The vertical strips $\Omega_k =\{\lambda\in\mathbb{C}|\,|r_k-{\rm Re}\,\lambda|\leq \delta r_k^p\}$, $r_k =(\alpha_{2k}+\alpha_{2k+1})/2$, belong to the resolvent set of $T$, provided that $\delta <b -b'$ and $ |k|\geq N$ for $N$ large enough. For $|k|\ge N+1$, denote by $\Pi_k$ the curvilinear trapezoid formed by the lines ${\rm Re}\,\lambda = r_{k-1}$, ${\rm Re}\,\lambda = r_{k}$, and the boundary of the parabola $P_{p,h}$. Assume that $Q_0$ is the Riesz projection corresponding to the (bounded) part of the spectrum of $T$ that lies outside $\bigcup_{|k|\geq N+1}\Pi_k$. And let $Q_k$, $|k|\ge N+1$, be the Riesz projection for the part of the spectrum of $T$ confined within $\Pi_k$. Main result of the work consists in proving that the system of the invariant subspaces $Q_k(H)$, $|k|\geq N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.
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