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On a conjugation and a linear operator (The research of geometric structures in quantum information based on Operator Theory and related topics)

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(1)42. 数理解析研究所講究録 第2033巻 2017年 42-49. On. conjugation. a. and. a. linear operator. by Muneo. Cho, Eungil Ko, Ji Eun Lee, Kôtarô Tanahashi. Abstract. 1. In this note,. jugations. on a. we. introduce the. complex. study of. some. classes of operators. concerning with. con‐. Hilbert space.. Definition. 2. Let \mathcal{H} be on. \mathcal{H}. .. For. a. \mathcal{L}(\mathcal{H}) be the set of all bounded linear operators $\sigma$(T) $\sigma$_{p}(T) $\sigma$_{a}(T) $\sigma$_{s}(T) $\sigma$_{\mathrm{e} (T) $\sigma$_{w}(T) be the spectrum, the. complex Hilbert. T\in \mathcal{L}(H). ,. let. space and ,. ,. ,. ,. ,. point spectrum, the approximate point spectrum, the surjective spectrum, the essential spectrum and the Weyl spectrum, respectively. Definition 1.. (1) (2). (1). For. T\in \mathcal{L}(\mathcal{H}). ,. we. define. $\alpha$_{rn}(T). and. $\alpha$_{m}(T)=\displaystyle \sum_{j=0}^{m}(-1)^{j} \left(\begin{ar ay}{l m\ j \end{ar ay}\right)T^{*m-j}T^{j}, $\beta$_{m}(T)=\displaystyle \sum_{j=0}^{m}(-1)^{j} \left(\begin{ar ay}{l m\ j \end{ar ay}\right)T^{*rn-j}T^{m-j}. T is said to be. (2) T is said \mathbb{C}:|z|=1\}.. to be. m. ‐symmetric if $\alpha$_{m}(T)=0. m ‐isometric. if. $\beta$_{m}(T)=0. .. .. $\beta$_{m}(T). as. follows;. Then. (-i)^{m-1}$\alpha$_{m-1}(T)\geq 0. Then. $\beta$_{m-1}(T)\geq 0. and. and. $\sigma$(T)\subset \mathbb{R}.. $\sigma$_{a}(T)\subset \mathrm{N}=\{z\in. It holds that. (1) T^{*}$\alpha$_{rn}(T)-$\alpha$_{m}(T)T=$\alpha$_{m+1}(T) Proposition. invertible,. 1. ,. (Proposition 1.23, [1]) (m-1) ‐isometric.. then T is. (2) T^{*}$\beta$_{m}(T)T-$\beta$_{\mathfrak{m}}(T)=$\beta$_{m+1}(T) Let T be. m ‐isometric.. If. m. is. .. even. and T is.

(2) 43. When. is. m. odd,. have the. following:. (Theorem 1, [7]) If m \dot{u} any which is not (m-1) ‐isometric.. Proposition m ‐isometric. we. Proposition. odd. 2. 3. (Theorem 3.4, [13]) IfT. is. m. number, then there. ‐symmetric and m\dot{u}. exists. an. invertible. even, then T is. (m-1)-. symmetric.. (1) (2) (3). Let T be. 1‐symmetric. Then $\tau$*-T=0 So T is Hermitian clearly. 2‐symmetric. By Proposition 3, T is 1‐symmetric. Hence T is Hermitian. be m‐symmetric. For sequences of unit vectors {xn}, {yn}, if (T-a)x_{n} \rightarrow .. Let T be Let T. (T-b)y_{n} \rightarrow 0(a\neq b) then \{x, y ) =0. and. If. \bullet. In. Q. is. [11],. 2‐nilpotent, then Q. is. ,. then. \{x_{n}, y_{n}\rangle. In. [1],. J. W. Helton introduced. J.. Agler and M.. .. Hence if. 0. Tx=ax, Ty=by(a\neq b). ,. 3‐symmetric. m. ‐symmetric for the study of Jordan operators.. If T is 1‐isometric, then T^{*}T-I=0 and T is. \bullet. 0. \rightarrow. Stankus studied. an. ‐isometric m_{ $\Gamma$}. isometry.. for the research of Dirichlet Differential. operators. We have many results of. Bermúdes,. 3. m‐isometric. Researchers. Conjugation. Definition 2. C. :. \mathcal{H}. \rightarrow. \mathcal{H} is said to be antilinear if. C(ax+by)=\overline{a}Cx+\overline{b} Cy,. for all a,. An antilinear operator C is said to be. C^{2}=I, \{ Cx, Cy\}=\{y, x\rangle \bullet. operators.. Martinón and etc.. If C is. a. a. b\in \mathbb{C}, x, y\in \mathcal{H}.. conjugation if. for all x,. y\in \mathcal{H}.. conjugation, then \Vert Cx\Vert=\Vert x\Vert for all x\in \mathcal{H}.. are. Agler, Stankus, Gu,.

(3) 44. Example. 4. Example. 1. Typical Example. of. Conjugation:. (1) J(z_{1}, z_{2}, z_{n})=(\overline{z_{1}},\overline{z_{2}}, \overline{z_{n}}). Let \mathcal{H}=\mathbb{C}^{n}.. (2) C(z_{1}, z_{2}, z_{n})=(\overline{z_{n}},\overline{z_{n-1}}, \ldots,\overline{z_{1}}). ,. .. J, C are conjugations. Example 2 Then. T is said to be. complex symmetric if there. Typical Example of. T=. a. .. a. conjugation C such that CTC= $\tau$*.. complex symmetric operator. \left(bgin{ary}l a_{0}& -1}&\cdots&a_{-(n1)}\ a_{1}& 0 \cdots&a_{-(n2)}\ &dots&\dots&\ a_{n-1}&a_{n-2}&\cdots&a_{0} \end{ary}\ight). Then CTC= $\tau$*. exists. Hence every. T : Let \mathcal{H}=\mathbb{C}^{n} and T be. (Toeplitz matrix).. Toeplitz. matrix is. complex symmetric ( C‐symmetric).. Takagi first showed this. He studied antilinear eigen‐value problem. following result. T.. Takagi. Factorization Theorem. Let T be. there eaist. a. symmetric and C ‐symmetric. matrix. Then. unitary U and normal and symmetric N such that T=UN{}^{t}U.. Symmetric operators. 5. [12]. S.. Jung, E. operators. We only In. Ko and J. E. Lee showed several results about set the. Theorem 1. Let C be. a. following. $\sigma$_{s}. (CTC) =\overline{$\sigma$_{s}(T)},. It is not need CTC= $\tau$*. .. $\sigma$_{\mathrm{p} (CTC) $\sigma$_{e}. complex symmetric. theorem.. conjugation and T\in \mathcal{L}(\mathcal{H}). $\sigma$(CTC)=\overline{ $\sigma$(T)},. \bullet. a. There is the. =\overline{$\sigma$_{p}(T)}. .. ,. (CTC) =\overline{$\sigma$_{e}(T)},. Then $\sigma$_{a}. (CTC) =\overline{$\sigma$_{a}(T)},. $\sigma$_{w}. (CTC) =\overline{$\sigma$_{w}(T)}.. It is the relation between spectra of T and CTC..

(4) 45. (m, C) ‐symmetric operator. 6. Deflnition 3. Let C be. a. conjugation and T\in \mathcal{L}(\mathcal{H}) Then .. $\Delta$_{m}(T;C)=\displaystyle\sum_{j=0}^{m}(-1)^{j}\left(\begin{ar ay}{l m\ j \end{ar ay}\right)T^{*j}CT^{m-j}C. (m, C) ‐symmetric ‐complex symmetric.) T is said to be. m. We have. if. $\Delta$_{m}(T;C). =. 0. (In [2]. .. T^{*}\cdot \mathrm{A}_{m}(T;C)-$\Delta$_{m}(T;C) (CTC) =$\Delta$_{m+1}(T;C). it is said to be. .. Hence if T is. (m, C) ‐symmetric,. then T is. (n, C) ‐symmetric. At the last year RIMS. symmetric. 7. [3],. and. means m. Conference, in [5] we already had ‐complex symmetric. Please see [5].. .. for every a. n(\geq m). talk of this class.. [m, C] ‐symmetric operator. Definition 3. Let C be. a. conjugation and T\in \mathcal{L}(\mathcal{H}) Then .. $\alpha$_{m}(T;C)=\displaystyle\sum_{j=0}^{m}(-1)^{j}\left(\begin{ar ay}{l} m\ j \end{ar ay}\right)(CT^{m-j}C)T^{j}. T is said to be. We have. [m, C] ‐symmetric. if. $\alpha$_{m}(T;C)=0.. CTC\cdot$\alpha$_{m}(T;C)-$\alpha$_{m}(T;C)\cdot T=$\alpha$_{m+1}(T;C). Hence if T is. [m, C] ‐symmetric,. Theorem 2. Let C be. then T is. .. [n, C] ‐symmetric. conjugation and T\in \mathcal{L}(\mathcal{H}) [m, C] ‐symmetriic if and only if so is $\tau$*. a. for every. n(\geq m). .. (a) (b) If T\dot{u}[m, C] ‐symmetric, then so is T^{n} for every n\in \mathbb{N}. (c) If T is [m, C] ‐symmetric and invertible, then T^{-1} is [m, C] ‐symmetric. T is. Theorem 3. Let T be. [m, C] ‐symmetric.. .. Then. $\sigma$(T)=\overline{ $\sigma$(T)}, $\sigma$_{p}(T)=\overline{$\sigma$_{p}(T)}, $\sigma$_{a}(T)=\overline{$\sigma$_{a}(T)}, $\sigma$_{s}(T)=\overline{$\sigma$_{s}(T)}.. .. (m, C)-.

(5) 46. \bullet. A pair. (T, S). is said to be C‐doubly. Lemma 1. Let. (T, S). be C ‐doubly. commuting if TS=ST and csc\cdot $\tau$=T CSC. .. commuting. Then. it holds. $\alpha$_{m}(T+S;C)=\displaystyle\sum_{j=0}^{m}\left(\begin{ar ay}{l m\ j \end{ar ay}\right)$\alpha$_{j}(T;C)\cdot$\alpha$_{m-j}(S;C). .. [m, C] ‐symmetric and S be [n, C] ‐symmetric. If (T, S) T+S\dot{u}[m+n-1, C] ‐symmetric.. Theorem 4. Let T be. commuting,. then. Theorem 4. Let. Q. be. n. ‐nilpotent. Then Q. is. [2n-1, C] ‐symmetric for. every. is C ‐doubly. conjugation. C. Theorem 5.. [m, C] ‐symmetric and Q be n ‐nilpotent. If (T, Q) T+Q is [m+2n-2, C] ‐symmetric.. Let T be. commuting, then Lemma 2. Let. (T, S). be C ‐doubly commuting. Then it holds. $\alpha$_{m}(TS;C)=\displaystyle \sum_{j=0}^{m}\left(\begin{ar ay}{l} m\ j \end{ar ay}\right)$\alpha$_{j}(T;C)\cdot T^{m-j}\cdot CS^{j}C\cdot$\alpha$_{m-j}(S;C) then. Theorem 7.. .. [m, C] ‐symmetric and S be [n, C] ‐symmetric. If (T, S) TS\dot{u}[m+n-1, C] ‐symmetric.. Theorem 6. Let T be. commuting,. is C ‐doubly. Let T be. [m, C] ‐symmetnc. [n, D] ‐symmetric.. and S be. is C ‐doubly. Then T\otimes S is. [m+n-1, C\otimes D] ‐symmetric. Proof.. It is clear that C\otimes D is. a. conjugation. on. \mathcal{H}\otimes \mathcal{H}. .. And it is easy to. see. that. [m, C\otimes D] ‐symmetric (T\otimes I, I\otimes S) is C\otimes D ‐doubly commuting. Since T\otimes S=(T\otimes I)(I\otimes S) by the previous theorem we have T\otimes S is [m+n-1, C\otimes D] ‐symmetric. Q.E.D. and I\otimes S is. T\otimes I is. [n, C\otimes D] ‐symmetric. Also it is clear that ,. 8. (m, C) ‐isometric operator. Definition 4. Let C be. a. conjugation and T\in \mathcal{L}(\mathcal{H}) Then .. $\Lambda$_{m}(T;C)=\displaystyle \sum_{j=0}^{m}(-1)^{j}\left(\begin{ar ay}{l} m\ j \end{ar ay}\right)T^{*m-j}(CT^{m-j}C). ..

(6) 47. T is said to be. We have. (m, C) ‐isometric. if. $\Lambda$_{m}(T;C)=0.. T^{*}\cdot$\Lambda$_{m}(T;C)\cdot(CTC)-$\Lambda$_{m}(T;C)=$\Lambda$_{m+1}(T;C). Hence if T is. (m, C) ‐isometric,. Theorem 8. Let T be. then T is. (n, C) ‐isometric. .. for every. n(\geq m). .. (m, C) ‐isometric. Then;. (a) T is bounded below, (b) 0\not\in$\sigma$_{a}(T) (c) T \dot{u} injective and R(T) ,. (d) if z\in$\sigma$_{a}(T) (e) if there. ,. then. is. \displaystyle\frac{1}{\oT^{-1} verline{z}\in$\sigma$_{a}(T^{*}). exists T^{-1} , then. Theorem 9. Let T be. closed, is. ,. (m, C) ‐isometric.. (m, C) ‐isometric. If $\tau$*. has. SVEP,. $\sigma$(T)=$\sigma$_{a}(T)=$\sigma$_{ $\epsilon$}(T) Theorem 10.. Let T be. normaloid, then. T is. \bullet. Of course, if T is. \bullet. A. pair (T, S). .. (m, C) ‐isometric. If T is power ( 1, C)-\dot{u} ometric, i. e., T^{*}CTC=I.. m‐isometric. and power. is said to be C-* doubly. Lemma 3. Let. then. (T, S). be C-* doubly. bounded,. bounded and T^{*}CTC-I is. then T is isometric.. commuting if TS=ST and S^{*}. CTC=CTC\cdot S^{*}.. commuting. Then. it holds. $\Lambda$_{m}(T+S;C)=\displaystyle\sum_{3m_{1}+m2+m=m}\left(\begin{ar ay}{l m\ m_{1},m_{2},m_{3} \end{ar ay}\right). (T^{*}+S^{*})^{m}1S^{*m_{2}}$\Lambda$_{m\mathrm{s}}(T;C)\cdot(CT^{m2}C)\cdot(CS^{ $\pi$ v1}C) It follows from the. .. following equation:. ((a+b)(c+d)-1)^{m}=((ac-1)+(a+b)d+bc)^{rn}. =\displaystyle\sum_{3m1+m2+m=m}\left(\begin{ar ay}{l m\ m_{1},m_{2},m_{3} \end{ar ay}\right). .. (a+b)^{m}b^{m}(ac-1)^{m3}c^{m}2d^{m1}..

(7) 48. Hence. we. have the. following. result.. (m, C) ‐isometric, Q be (m+2n-2, C) ‐isometric.. Theorem 11. Let T be. pair. Then. T+Q. Lemma 4. Let. is. (T, S). n. ‐nilpotent and (T, Q). be. commuting. be C-* doubly commuting. Then it holds. $\Lambda$_{m}(TS;C)=\displaystyle\sum_{j=0}^{m}\left(\begin{ar ay}{l m\ j \end{ar ay}\right)T^{*j}\cdot$\Lambda$_{m-\mathrm{j} (T;C)(CT^{j}C)\cdot$\Lambda$_{j}(S;C). It follows from the. a. .. following equation:. (abcd-1)^{m}=((ab-1)+a(cd-1)b)^{m}. Hence. we. have the. =\displaystle\sum_{j=0}^{m}\left(\begin{ar y}{l m\ j \end{ar y}\right). following. Theorem 12. LetT be. commuting, then TS is Theorem 13.. .. a^{j}(ab-1)^{m-j}b'(cd-1)^{j}.. result.. (m, C) ‐isometric andS be (n, C) ‐isometric. If (T, S) (m+n-1, C) ‐isometric.. Let T be. (m, C) ‐isometric. and S be. (n, D) ‐isometric.. is C-* doubly. Then T\otimes S is. (m+n-1, C\otimes D) ‐isometric.. (m, C\otimes D) ‐isometric and I\otimes S is (n, C\otimes D) ‐isometric. Also it is clear that (T\otimes I, I\otimes S) is C\otimes D-* doubly commuting. Since T\otimes S=(T\otimes I)(I\otimes S) by the previous theorem we have T\otimes S is (m+n-1, C\otimes D) ‐isometric. Q.E.D. Proof.. It is easy to. see. that T\otimes I is. ,. References [1]. Agler and M. Stankus, m‐Isometric transformations of Hilbert space I, Integr. Equat. Oper. Theory, 21(1995) 383‐429. [2] M. Chō, E. Ko and J. E. Lee, On m‐complex symmetric operators, Mediterr. J. Math. J.. 13(2016) 2025‐2038. [3] M. Cho, E. Ko and J. E. Lee, On m‐complex symmetric operators II, Mediterr. J. Math. 13(2016) 3255‐3264. [4] M. Cho, E. Ko and J. E. Lee, On (m, C) ‐isometric operators, Complex Anal. Oper. Theory, 4(2016) 1679‐1694. [5] M. Cho‐, E. Ko and J. E. Lee, On m‐complex symmetric operators, RIMS Kôkyûroku.

(8) 49. 1996. [6] [7]. M. M.. (2016) April, Cho, Cho,. 23‐33.. Tanahashi, On [m, C] ‐symmetric operators, preprint. Tanahashi, Invertible weighted shift operators which are. J. E. Lee and K. S.. Ôta. and K.. isometries, Proc. Amer. Math. Soc. 141(2013). [8]. C. Gu and M.. Products and. [9] [10]. Soc.. [11]. 358(2005). Some results. on. with. 1285‐1315.. S. R. Garcia and M.. 359(2007). 11701‐6.. Stankus, higher order isometries and symmetries: a nilpotent operators, Linear Algebra Appl., 469(2015) 500‐509. and M. Putinar, Complex symmetric operators, Trans. Amer. Math.. sums. S. R. Garcia. Soc.. m‐. Putinar, Complex symmetric operators II,. Trans. Amer. Math.. 3913‐3931.. J. W.. Helton, Infinite dimensional Jordan operators and Strum‐Liouville conjugate point theory, Trans. Amer. Math. Soc. 170(1972) 305‐331. [12] S. Jung, E. Ko and J. E. Lee, On complex symmetric operators, J. Math. Anal. Appl. 406 (2013) 373‐385. [13] S. McCullough and L. Rodman, Hereditary classes of operators and matrices, Amer. Math. Monthly, 104(1997) 415‐430. Muneo Ch6. Department of Mathematics, Kanagawa University, Hiratsuka 259‐1293, Japan \mathrm{e} ‐mail:. chiyom01@kanagawa‐u.ac.jp. Eungil Ko Department of Mathematics, Ewha Womans University, Seoul 120‐750, \mathrm{e} ‐mail:. Korea. eiko@ewha.ac.kr. Ji Eun Lee. Department of Mathematics‐Applied Statistics, Sejong University, Seoul 143‐747, Korea ‐mail:. jieun7@ewhain.net; jieunlee7@sejong.ac.kr. Kôtarô Tanahashi. Department 8558, Japan \mathrm{e}. ‐mail:. of Mathematics, Tohoku Medical and Pharmaceutical. tanahasi@tohoku‐mpu.ac.jp. University, Sendai. 981‐.

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