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(1)32. 数理解析研究所講究録 第2033巻 2017年 32-41. Complex symmetric operators. and their. Weyl. type theorems Il Ju. An, Eungil Ko,. and Ji Eun Lee. Abstract. study. We. a. necessary and sufficient condition for. complex symmetric. operator matrices to satisfy a‐Weyl’s theorem. Moreover, we also give the conditions for such operator matrices to satisfy generalized a‐Weyl’s theorem and generalized a ‐Browder’s theorem, respectively. As some applications, we provide various examples of such operator matrices which satisfy Weyl type. theorems.. Introduction. 1. Let \mathcal{H} be. the. an. infinite dimensional. separable Hilbert. $\sigma$(T). ,. \mathcal{L}(\mathcal{H}). space and let. of bounded linear operators acting on \mathcal{H} If T \in \mathcal{L}(\mathcal{H}) , and for the the spectrum, point spectrum, the $\sigma$_{a}(T) $\sigma$_{p}(T) , $\sigma$_{s}(T) ,. algebra. .. denote. we. write. surjective. spectrum, and the approximate point spectrum of T respectively. ,. T\in \mathcal{L}(\mathcal{H}). N(T) and R(T) for the null space and the range of $\alpha$(T):=dimN(T) and $\beta$(T) :=dimN(T^{*}) respectively. For T \in \mathcal{L}(\mathcal{H}) the smallest nonnegative integer p such that N(T^{p}) N(T^{p+1}) is called the ascent of T and denoted by p(T) If no such integer exists, we set \infty The smallest nonnegative integer q such that R(T^{q}) p(T) R(T^{q+1}) is called the descent of T and denoted by q(T) If no such integer exists, we set q(T)=\infty. If. T,. ,. we. shall write. respectively. Also,. let. ,. =. ,. .. =. =. .. .. A conjugation. \{Cx, Cy\rangle = \langle y, x\}. on. \mathcal{H} is. an. antilinear operator C : \mathcal{H} I For any. for all x, y \in \mathcal{H} and C^{2}. =. .. \rightarrow. \mathcal{H} which satisfies. conjugation C there ,. 02010 Mathematics Subject Classification; Primary 47\mathrm{A}10, 47\mathrm{A}53, 47\mathrm{A}55. This research was. supported by Basic Science Research Program through the National Research funded by the Ministry of Education (2014\mathrm{R}\mathrm{l}\mathrm{A}\mathrm{l}\mathrm{A}2056642) This re‐ search was supported by Basic Science Research Program through the National Research Founda‐ tion of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013006537). The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Tech‐ nology (2016\mathrm{R}\mathrm{l}\mathrm{A}2\mathrm{B}4007035) Foundation of. Korea(NRF). .. ..

(2) 33. is. an. orthonormal basis. \{e_{n}\}_{n=0}^{\infty}. Ce_{n}. for \mathcal{H} such that. =. e_{n} for all. n. (see [7]. for. details). An operator T \in \mathcal{L}(\mathcal{H}) is said to be complex symmetric if there exists a conjugation C on \mathcal{H} such that T=CT^{*}C In this case, we say that T is more. .. complex symmetric with conjugation C This concept is due to the fact that T is a complex symmetric operator if and only if it is unitarily equivalent to a symmetric matrix with complex entries, regarded as an operator acting on an l^{2} ‐space of the appropriate dimension (see [7]). All normal operators, Hankel matrices, finite Toeplitz matrices, all truncated Toeplitz operators, and some Volterra integration operators are included in the class of complex symmetric operators. We refer the .. reader to. [7]-[9]. for. more. details.. Weyl type theorems for upper triangular operator matrices have been studied by many authors. In general, even though Weyl type theorems hold for The. \left(\begin{ar y}{l T_{1}&0\ 0&T_{2} \end{ar y}\right) (see [10], [11], [13], [14], [3],. entry operators T_{1} and T_{2} neither. nor. ,. theorems. and ect. the relation between. matrix and. diagonal of Weyl type theorems. Recently, symmetric operator a‐Browder’s now. matrices. theorem for. in. some. satisfies. Weyl type. So many authors have been studied. triangular operator matrix [17], they provide several forms of complex. \left(\begin{ar y}{l T_{1}&T_{2}\ T_{3}&T_{4} \end{ar y}\right). an. upper. and have studied. complex symmetric operator. a‐Weyl’s. matrices. consider how. when. theorem and. \left(\begin{ar y}{l A&B\ 0&CA^{*}C \end{ar y}\right). .. We. Weyl type theorems hold for upper triangular operator matrices entry operators are complex symmetric.. In this paper, B is. a. \left(\begin{ar y}{l T_{1}&T_{3}\ 0&T_{2} \end{ar y}\right). we. focus. complex symmetric. on. the operator matrix. with the. which the operator matrix. conjugation C. \left(\begin{ar y}{l A&B\ 0&CA^{*}C \end{ar y}\right). .. satisfies. \left(\begin{ar y}{l A&B\ 0&CA^{*}C \end{ar y}\right) \in\mathcal{L}(\mathcal{H}\oplus\mathcal{H}). In this case,. we are. when. interested in. Weyl type theorems under what. behavior of the entry operator A In particular, we give a necessary and sufficient condition for this complex symmetric operator matrices to satisfy a‐Weyl’s theo‐ .. Moreover, we also provide the conditions for such operator matrices to satisfy generalized a‐Weyl’s theorem and generalized a‐Browder’s theorem, respectively. As some applications, we give various examples of such operator matrices which satisfy Weyl type theorems. rem.. 2. Preliminaries. An operator T \in \mathcal{L}(\mathcal{H}) is called upper semi‐Fredholm if it has closed range and finite dimensional null space and is called lower semi‐Fredholm if it has closed range. and its range has finite co‐dimension. If T\in \mathcal{L}(\mathcal{H}) is either upper or lower semi‐ Fredholm, then T is called semi‐Fredholm, and index of a semi‐Fredholm operator.

(3) 34. T\in \mathcal{L}(\mathcal{H}). is defined. by. i(T):= $\alpha$(T)- $\beta$(T) $\alpha$(T). If both. is called. $\beta$(T). and. Weyl. are. finite, then. .. T is called Fredholm. An operator T\in \mathcal{L}(\mathcal{H}) zero and Browder if it is Fredholm of. if it is Fredholm of index. finite ascent and. descent, respectively. The left essential spectrum $\sigma$_{SF+}(T) the right essential spectrum $\sigma$_{SF-}(T) the essential spectrum $\sigma$_{e}(T) the Weyl spectrum $\sigma$_{w}(T) and the Browder spectrum $\sigma$_{b}(T) of T\in \mathcal{L}(\mathcal{H}) are defined \Re follows; ,. ,. ,. ,. $\sigma$_{SF+}(T). :=. { $\lambda$\in \mathbb{C}. :. T- $\lambda$ is not upper. semi‐Fredholm},. :=. { $\lambda$\in \mathbb{C}. :. T- $\lambda$ is not lower. semi‐Fredholm},. $\sigma$_{SF-}(T) $\sigma$_{e}(T). :=. $\sigma$_{w}(T). { $\lambda$\in \mathbb{C}. :=. Fredholm},. T- $\lambda$ is not. :. { $\lambda$\in \mathbb{C}. :. T- $\lambda$ is not. Weyl},. T- $\lambda$ is not. Browder},. and. $\sigma$_{b}(T). :=. { $\lambda$\in \mathbb{C}. :. respectively. Evidently. $\sigma$_{SF+}(T)\cup$\sigma$_{SF-}(T)=$\sigma$_{e}(T)\subseteq$\sigma$_{w}(T) \subseteq$\sigma$_{b}(T)=$\sigma$_{e}(T)\cup acc $\sigma$(T) where. $\Delta$\backslash. we. acc. write. acc. $\Delta$ then ,. we. $\Delta$ for the accumulation points of $\Delta$\subseteq \mathbb{C}. .. If. we. write iso $\Delta$=. let. $\pi$_{00}(T). :=. { $\lambda$\in. iso. $\sigma$(T). :. 0< $\alpha$(T- $\lambda$)<\infty },. p_{00}(T) := $\sigma$(T)\backslash $\sigma$_{b}(T) We say that Weyl’s theorem holds for $\sigma$(T)\backslash $\sigma$_{w}(T) $\pi$_{00}(T) and that Browder’s theorem holds for T $\sigma$(T)\backslash $\sigma$_{w}(T)=p_{00}(T) We recall the definitions of some spectra;. and. .. if. =. ,. .. $\sigma$_{ $\epsilon$ a}(T):=\cap\{$\sigma$_{a}(T+K):K\in \mathcal{K}(\mathcal{H})\} is the essential. approximate point spectrum, and. $\sigma$_{ab}(T). :=\cap. is the Browder essential. $\pi$_{00}^{a}(T) and. { $\sigma$_{a}(T+K):TK=KT. T\in \mathcal{L}(\mathcal{H}). .. and. approximate point spectrum.. :=. { $\lambda$\in. p_{00}^{a}(T)=$\sigma$_{a}(T)\backslash $\sigma$_{ab}(T). Let. ,. iso. $\sigma$_{a}(T). :. K\in \mathcal{K}(\mathcal{H}) }. We put. 0< $\alpha$(T- $\lambda$)<\infty}. .. We say that a‐Browder’s theorem holds. $\sigma$_{a}(T)\backslash $\sigma$_{ea}(T)=p_{00}^{a}(T). ,. for T if. \mathcal{L}(\mathcal{H}) \mathcal{L}(\mathcal{H}) if. T \in \in.

(4) 35. and. a‐. Weyl’s theorem holds for T. if. $\sigma$_{a}(T)\backslash $\sigma$_{ea}(T)=$\pi$_{00}^{a}(T). .. It is known that. a‐Weyl’s. theorem. a‐Weyl’s Let. ‐Browder’s theorem. \vec{\frac{}{}} a. theorem. T_{n}=T|_{\mathrm{R}(T^{n})}. \Rightar ow \mathrm{W}\mathrm{e}\mathrm{y}\mathrm{l} ’s. for each. theorem. \Rightarrow. \vec{\frac{}{}}. nonnegative integer. Browder’s. theorem,. Browder’s theorem.. n| in. particular, T_{0}=T. .. If. T_{n} is. upper semi‐Fredholm for some nonnegative integer n then T is called a upper semi‐ B ‐Predholm operator. In this case, by [4], T_{m} is a upper semi‐Fredholm operator and ind(T_{rn}) =ind(T_{n}) for each m \geq n Thus, we can consider the index of T ,. .. the index of the semi‐Fredholm operator T_{n} Similarly, we define lower semi‐B‐ Fredholm operators. We say that T\in \mathcal{L}(\mathcal{H}) is B ‐Fredholm if it is both upper and. as. .. lower semi‐B‐Fredholm. Let. operators such that. SBF_{\overline{+} (\mathcal{H}) be ind(T)\leq 0 and let. the class of all upper semi‐B‐Fredholm. ,. $\sigma$_{\mathcal{S}BF_{+}^{-} (T):=\{ $\lambda$\in \mathbb{C}:T- $\lambda$\not\in SBF_{+}^{-}(\mathcal{H})\}. An operator T \in B‐ Weyl spectrum. \mathcal{L}(\mathcal{H}) is $\sigma$_{BW}(T). $\sigma$_{BW}(T) In. addition,. we. :=. state two. called B‐. Weyl if it by. is B ‐Fredholm of index. zero.. The. of T is defined. { $\lambda$\in \mathbb{C}. :. spectra. T- $\lambda$ is not as. a. B ‐Weyl. operator}.. follows;. $\sigma$_{LD}(T)=\{ $\lambda$\in \mathbb{C}|T- $\lambda$\not\in LD(\mathcal{H})\}, $\sigma$_{RD}(T)=\{ $\lambda$\in \mathbb{C}|T- $\lambda$\not\in RD(\mathcal{H})\}, where. LD(\mathcal{H}) \mathcal{H}| q(T). {T. \mathcal{H}| p(T). R(\mathcal{I}^{ $\varphi$(T)+1}). closed}, and RD(\mathcal{H}) \infty and is T\in < The notation { R(T^{q(T)}) closed}. p_{0}(T) (respectively, p_{0}^{a}(T)) denotes the set of all poles (respectively, left poles) of T while $\pi$_{0}(T) (re‐ spectively, $\pi$_{0}^{a}(T) ) is the set of all eigenvalues of T which is an isolated point in $\sigma$(T) (respectively, $\sigma$_{a}(T) ). =. \in. < \infty. and. is. =. ,. Let. (i). T\in \mathcal{L}(\mathcal{H}). T satisfies. (ii) (iii) (iv). .. We say that. generalized Browder‘s theorem. T satisfies. T satisfies T satisfies. if. generalized a‐Browder’s theorem. $\sigma$(T)\backslash $\sigma$_{BW}(T)=p_{0}(T) ; if. $\sigma$_{a}(T)\backslash $\sigma$_{SBF_{+}^{-} (T)=p_{0}^{a}(T) ;. generalized Weyl’s theorem if $\sigma$(T)\backslash $\sigma$_{BW}(T)=$\pi$_{0}(T) ; generalized a‐ Weyl’s theorem if $\sigma$_{a}(T)\backslash $\sigma$_{SBF_{+}^{-} (T)=$\pi$_{0}^{a}(T). It is known that. generalized a‐Weyl’s theorem. \Rightarrow. generalized Weyl’s theorem. ..

(5) 36. \Downar ow generalized. \Downar ow. a‐Browder’s. An operator T\in. \mathcal{L}(\mathcal{H}). theorem. has the. generalized Browder’s theorem.. \Rightarrow. single‐valued. extension. property. at. for every open neighborhood U of $\lambda$_{0} the only analytic function f which satisfies the equation (T- $\lambda$)f( $\lambda$)=0 is the constant function. $\lambda$_{0} U. :. \in. \mathb {C} if. \mathcal{H}. \rightarrow. f\equiv 0. on. U.. The operator T is said to have the single‐valued extension property if T has the single‐valued extension property at every $\lambda$_{0}\in \mathbb{C}.. In this. trices. study Weyl type theorems for complex symmetric operator [17], they provide several forms of complex symmetric operator ma‐. section,. matrices. In. \left(begin{ar y}{l T_{\mathrm{l}&T_{2}\ T_{3}&T_{4} \end{ar y}\right). symmetric with. we. matrix .. \left(\begin{ar y}{l s*&0\ 0&S \end{ar y}\right). They. if C is. Indeed,. .. \left(\begin{ar y}{l C&0\ 0&C \end{ar y}\right). conjugation C and T_{4}. \mathcal{H}. Theorem. Wyel Type. 3. if and. does not. matrices. for any set $\Delta$ in \mathb {C} Lemma 3.1. .. For. example,. \mathcal{H} , then are. the. ‐Weyl. a. where C is our. ([17]) If C\dot{u}. study, a. a. we. .. conjugation. on. \mathcal{H} and. =. ,. ,. Remark that if S is. is the unilateral shift. we. \mathcal{H}. on. complex. study generalized. Put $\Delta$^{*}. .. following. A\in \mathcal{L}(\mathcal{H}). (i) $\sigma$(A)^{*} $\sigma$(CAC) $\sigma$_{p}(A)^{*} $\sigma$_{p}(CAC) $\sigma$_{a}(A)^{*} $\sigma$_{s}(CAC)^{*}. (ii) $\sigma$_{e}(A)^{*}=$\sigma$_{e}(CAC) and $\sigma$_{w}(A)^{*}=$\sigma$_{w}(CAC) ,. a. complex symmetric operator. start with the. on. complex. complex symmetric with. In this paper,. identities hold: =. is. complex symmetric operator. theorem for. conjugation. \left(begin{ar y}{l T_{\mathrm{l}&T_{2}\ T_{3}&T_{4} \end{ar y}\right). and a‐Browder’s theorem for. \left(\begin{ar y}{l T_{1}&T_{2}\ 0&CT_{\mathrm{l}^{*}C \end{ar y}\right). generalized. \left(\begin{ar y}{l T_{\mathrm{l} &T_{2}\ T_{3}&CT_{1}^*C \end{ar y}\right). For. T_{2} and T3. a‐Weyl’s theorem. symmetric operator matrices and. .. if. on. satisfy Weyl’s theorem where S. also have studied. Weyl theorem. only. CT_{1}^{*}C. =. conjugation. a. =. :=. \{\overline{z}: z\in $\Delta$\}. lemmas.. ,. then the. $\sigma$_{a}(CAC). ,. and. following. $\sigma$_{s}(A). =. .. complex symmetric operator with the conjugation C then 3.5] that S has the single‐valued extension property if and only if S^{*} has. With the similar proof of [16], we have the following lemma. it is known from. [16,. a. ,. Lemma. Lemma 3.2 Let C be. a. valued extension property. conjugation. on. \mathcal{H} and. S\in \mathcal{L}(\mathcal{H}). if and only if CSC has.. .. Then S has the. single‐.

(6) 37. Lemma 3.3. If C. is. a. conjugation. on. \mathcal{H} and. A\in \mathcal{L}(\mathcal{H}). ,. then the. following. iden‐. tities hold:. (i) $\sigma$_{b}(A)^{*}=$\sigma$_{b}(CAC) and $\sigma$_{D}(A)^{*}=$\sigma$_{D} (CAC). (ii) $\sigma$_{LD}(A)^{*}=$\sigma$_{LD} (CAC) and $\sigma$_{RD}(A)=$\sigma$_{RD}(CAC)^{*}. (iii) $\sigma$_{BF}(A)^{*}=$\sigma$_{BF} (CAC) and $\sigma$_{BW}(A)^{*}=$\sigma$_{BW} (CAC). this paper,. Throughout put. M(A, B). =. conjugation C. }.. for operators A, B\in \mathcal{L}(\mathcal{H}). \{ \left(\begin{ar y}{l A&B\ 0&CA^{*}C \end{ar y}\right) We. the. single‐valued. (a). Then the. Let. us. \mathcal{L}(\mathcal{H}\oplus \mathcal{H}). study a‐Weyl. complex symmetric operator Theorem 3.4 Let. \in. B is. theorem and. matrices in. M(A, B). T\in M(A, B) Suppose .. extension. :. and. a. conjugation C. complex symmetric. generalized a‐Weyl. on. \mathcal{H},. with the. theorem for. .. that A is. complex symmetric which has. property.. following statements are equivalent; satisfies Weylfs theorem. (i) (ii) A satisfies a‐Weyl’s theorem. (iii) T satisfies Weyl’s theorem. (iv) T satisfies a‐Weyl’s theorem. (b) Then the following statements are equivalent; (i) A satisfies generalized Weylfs theorem. (ii) A satisfies generalized a‐Weyl’s theorem. (iii) T satisfies generalized Weyl theorem. (vi) T satisfies generalized a‐Weyl theorem. A. recall that the Hilbert. analytic functions f. Hardy. For any. on. where. ,. ,. the. consists of all. representation. \displaystyle\sum_{n=0}^{\infty}|a_{n}|^{2}<\infty.. H^{2}=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{z^{n}:n=0,1,2,3,\cdots\}}.. $\varphi$\in L^{\infty}. by H^{2}. the open unit disk \mathrm{D} with the power series. f(z)=\displaystyle \sum_{n=0}^{\infty}a_{n}z^{n} It is clear that. space, denoted. Toeplitz operator T_{ $\varphi$} : H^{2}\rightarrow H^{2}. is defined. by. the formula. T_{ $\varphi$}f=P( $\varphi$ f) for. f\in H^{2}. C_{2}. be the. where P denotes the. conjugations. on. orthogonal projection. of L^{2} onto H^{2}. H^{2} given by. (C_{1}f)(z)=\overline{f(\overline{z})} and (C_{2}f)(z)=\overline{f}(-\overline{z}) for all. f\in H^{2} respectively. ,. .. Let. C_{1} and.

(7) 38. 3.5 Let. Corollary and. C_{1}. in. \mathcal{L}(H^{2}\oplus H^{2}). C_{2} be all. the. conjugations. f\in H^{2} Suppose .. (^{$\tau$_{0} $\varphi$ c_{1}$\tau$_{$\varphi$} \tau$_{$\psi$}*c_{1}). T=. are. and. (C_{2}f)(z)=f(-\overline{z}) for. or. T=. on. H^{2} given by. that. (C_{1}f)(z)=\overline{f(\overline{z})}. \left(\begin{ar y}{l T_{$\psi$}&T_{$\varphi$}\ 0&C_{2}T_{$\psi$}^{*C_{2} \end{ar y}\right). where. \left\{ begin{ar ay}{l $\varphi$(z)=$\varphi$_{0}+2\sum_{k=1}^{\infty}\hat{$\varphi$}(2k)Re\{z^2k}\+2i\sum_{k=1}^{\infty}\hat{$\varphi$}(2k-1)Im\{z^2k-1}\ \ $\psi$(z)=$\psi$_{0}+2\sum_{n=1}^{\infty}\hat{$\psi$}(n)Re\{z^n}\. \end{ar ay}\right. If T_{ $\varphi$}. or. T_{ $\psi$}. have the. single‐valued. extension. (1). property, then T satisfies a‐Weyl’s. theorem.. Example. 3.6 Let C be. let U_{1} and. U_{2}. satisfies. are. conjugation. a. bilateral shifts. on. on. l^{2}(\mathbb{Z}). a‐Weyl’s theorem from Theorem. .. l^{2}(\mathbb{Z}) given by Then. 3.4.. Cx=\overline{x} for all. Corollary. 3.8 Let. proof of Theorem. 3.4 and. [18,. Theorem. T\in M(A, B) If A .. valued extension property, then the (i) A satisfies Browder’s theorem.. and. \left(\begin{ar y}{l U_{1}&U_{2}\ 0&CU_{\mathrm{l}^{*C \end{ar y}\right) \in \mathcal{L}(l^{2}(\mathbb{Z})\oplus l^{2}(\mathbb{Z}). Corollary 3.7 Let T\in M(N, B) where N is normal and B=CB^{*}C for jugation C. Then T satisfies generalized a‐ Weyl theorem. From the similar way with the get the following corollary.. x. is complex symmetric which has following statements are equivalent;. a con‐. 4.6],. the. we. single‐. (ii) A satisfies a‐Browder’s theorem. (iii) A satisfies generalized Browder’s theorem. (iv) A satisfies generalized a‐Browder’s theorem. (v) T satisfies Browder’s theorem. (vi) T satisfies a‐Browderfs theorem. (vii) T satisfies generalized Browder’s theorem. (viii) T satisfies generalized a‐Browder’s theorem. Recall that is. an. an. eigenvalue. operator T. of T. .. In. \in. is said to be isoloid if every $\lambda$ \in \mathrm{i}\mathrm{s}\mathrm{o} $\sigma$(T) proved that if T \in M(A, B) where A and. \mathcal{L}(\mathcal{H}). [17], they. isoloid operators with the single‐valued extension property and if Weyl’s theorem holds for both A and A^{*} , then a‐Weyl’s theorem holds for T Finally, we A^{*}. are. .. consider not. complex symmetric complex symmetric.. operator matrices where main. diagonal operators. are.

(8) 39. Theorem 3.9 Let. T\in M(A, B). where A and A^{*} have the. property. Then the following statements hold: (a) If A satisfies generalized Weyl theorem, then T. single‐valued. extension. satisfies generalized a ‐Weyl. theorem.. (b) If A. isoloid, then the following statements satisfies generalized Weyl theorem. (i) A satisfies generalized a‐ Weyl theorem. (ii) T satisfies generalized Weyl theorem. (iii) (iv) T satisfies generalized a‐Weyl theorem. (c) If A \dot{u} isoloid, then the following statements A and A^{*} satisfies Weyl’s theorem. (i) T satisfies Weyl’s theorem. (ii) (iii) T satisfies a‐Weyl theorem. is. are. equivalent;. A. are. equivalent;. Corollary 3.10 Let T\in M(A, N) where A is decomposable and N is normal or nilpotent of order 2 with N=CN^{*}C If A satisfies generalized Weyl’s theorem, then T satisfies generalized a ‐Weyl’s theorem. .. Example. 3.11 For x\in \mathbb{C}^{n} , define. Then \mathcal{C} is. a. S=\oplus_{j=1}^{\infty}S_{j}. conjugation. C^{j}(\displaystyle \sum_{i=1}^{n}$\alpha$_{i}e_{i})=\sum_{i=1}^{n}\overline{$\alpha$_{i} e_{n-i+1}. \mathcal{H} where dim\mathcal{H}. =. an. Then S is. n_{j}-1 weighted shift. \aleph_{0} Suppose. on. S_{j}=\left(bgin{ary}l 0&$\lambd_{athrml}^{(j)&0\cdots&0 &$\lambd_{2}^(j)&\cdots0 \cdots&0 \dots&0 &0 $\lambd_{nj}-\mathr{l^(j)}\ &0 \end{ary}ight). orthonormal basis of. complex symmetric \mathcal{H} defined. S_{j}. with. |$\lambda$_{k}^{(j)}| [23,. with C from. by. =. |$\lambda$_{n_{j}-k}^{(j)}|. Theorem. W= (x_{1}, x_{2}, x_{3}, \displaystyle \cdot \cdot ):=(\frac{1}{2}x_{2}, \frac{1}{3}x_{3}, \frac{1}{4}x_{4}, \cdot \cdot ) If T=. Indeed,. .. Put. C=\oplus C^{j}.. that S is written. .. as. where. with respect to .. on. }(\mathcal{H}\oplus \mathcal{H}) \left(\begin{ar y}{l W^{*}&S\ $\sigma$(W^{* 0&CWC \end{ar y}\righ}t)) \in\mathcal{L$\sigma$_{BW}(W^{* }) .. since. =. satisfies. Then T satisfies =. theorem.. generalized Weyl’s the single‐valued extension property. theorem from Theorem 3.9.. \{0\}. and. 3.1].. Let W be. =. a. .. generalized a‐Weyl’s. $\pi$_{0}(W^{*}). Moreover,. for all 1 \leq k \leq. theorem.. \emptyset it follows that W^{*} ,. in this case, W and W^{*} have. Hence T satisfies the. generalized a‐Weyl’s.

(9) 40. References [1] [2] [3]. P.. Aiena, Fredholm and local spectral theory with applications. multipliers,. to. Kluwer Academic Pub. 2004. P.. Aiena, M.T. Biondi. and C. Carpintero, On Drazin invertibility, Proc. (2008), no. 8, 2839‐2848.. Amer. Math. Soc. 136. An, Weyl type theorems for. I. J.. 2\times 2 operator. matrices, Kyung Hee Univ.,. Ph.D. Thesis. 2013.. [4]. M.. Berkani, On. a. class. of quasi‐Fredholm operators,. Int.. 244‐249.. [5]. S. R. Th.. [6] [7] [8]. Garcia, Aluthge transforms of complex symmetric operators,. 60(2008),. Math. Anal.. of unitaries, conjugations, Appl. 335(2007), 941‐947.. S. R. Garcia and M.. 358(2006),. 359(2007), Some. new. 362(2010),. Math. Soc.. and. applications,. 1285‐1315.. Complex symmetric operators. —,. Eq. Op.. and the Friedrichs operator, J.. Putinar, Complex symmetric operators. Trans. Amer. Math. Soc. —,. Int.. 357‐367.. Means. —,. Math. Soc.. [9]. Eq. Op. Th. 34(1999),. and. applications II,. Trans. Amer.. of complex symmetntc operators,. Trans. Amer.. 3913‐3931. classes. 6065‐6077.. [10]. S. V.. [11]. J. K.. Han, H.Y. Lee and W.Y. Lee, Invertible completions of 2 triangular operator matrtces, Proc. Amer. Math. Soc. 128 (1999),. 2 upper 119‐123.. [12]. I. S.. triangular. Djordjevic and Y. M. Han, A note on Weyl’s theorem for operator matrices, Proc. Amer. Math. Soc. 130(2003), 2543‐2547.. Hwang. and W.Y.. operator matrices,. [13] [14]. W.Y.. Lee,. 129(2001), —,. Int.. Lee, The boundedness below of 2\times 2 Eq. Op. Th. 39 (2001), 267‐276.. upper. Weyl spectra of operator matrices, Proc. Amer.. \times. Math.. Soc.. 131‐138.. Weyl’s theorem for operator matrices,. Int.. Eq. Op.. Th. 32. (1998),. 319‐331.. [15]. S.. Jung, E. Ko, M. Lee, and J. Lee, On local spectral properties of complex symmetric operators, J. Math. Anal. Appl. 379(2011), 325‐333..

(10) 41. [16]. [17] [18]. S.. Jung, E. Ko, and J. Lee, On scalar extensions and spectral decompositions of complex symmetric operators, J. Math. Anal. Appl. 382(2011), 252‐260. On. —,. 406(2013),. complex symmetric operator matrices,. —,. Properties of complex symmetric operator matrices, Operators. 8(4)(2014),. 957‐974.. [19]. E. Ko and J.. [20]. K. Laursen and M.. Neumann, An introduction. don. 2000.. Lee, On complex symmetric Toeplitz operators, Appl. 434(2016), 20‐34.. D.. Press, Oxford,. 1(2007),. Zhang, preprint.. [23]. S. Zhu and C. G.. S.. H.. Zhang,. and J.. Matri‐. Wu, Spectra of upper‐triangular operator matrix,. Li, Complex symmetric weigthed shifts,. Soc., 365(1)(2013),. 511‐530.. Il Ju An. Department of Mathematics Hankuk University of Foreign Studies Yongin‐si Gyeonggi‐do 17035 Korea. 66431004@naver.com. Eungil Ko Department of. Mathematics. Ewha Womans. University. Seoul 03760 Korea ‐mail: eiko@ewha.ac.kr Ji Eun Lee. Department of Mathematics‐Applied Statistics Sejong University Seoul 05006 Korea \mathrm{e}‐mail:. spectral theory, Claren‐. 419‐526.. [22]. \mathrm{e} ‐mail:. to local. J. Math. Anal.. Sarason, Algebraic properties of truncated Toeplitz operators, Oper.. ces,. Appl.. 373‐385.. and matrices.. [21]. J. Math. Anal.. jieun7@ewhain.net: jieunlee7@sejong.ac.kr. Trans. Amer. Math..

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This spectral triple encodes the kinematics of quantum gravity: the holonomy loops generate the algebra; the corresponding vector fields are packed in the Dirac type operator and