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z)c%M4>lHdD^IR+D(*kMR1|V)gi5y`qmY({{zV6#!R#@EIc4=fF0I5v1W}Uf^Q_Y%VK4ZXo#NAH0W@;ltAk0WzlZlWReOO%tQ$i;PFTRoY zZZdB_ZKgh$$O4wewVOqO&6xOfY`WdT!A+_$Tv7V|dBxb~f-V;KtQlZxSSj`W+iI8I zBXCcJ`s8`v)&-rF@t1~`kB6=%4)pG`Wr%xF2{8uA#xlIr+lqhxJsi zGG(v5EL_0s-p;h+qCE=GM<^V!uy&-r!wct5d@TjSuP|FKqEBRTB7_plR6!P}LpZT8 z^6!LUS@<)i0P;Z|(zDYGXGIK?hLv@S&c0QQm1deSjbc?4#ya{&9l6x}M)b~hJ2Br*^ir1u*Xf|sxW diff --git a/euclidean-spaces.tex b/euclidean-spaces.tex index 46c7477..71e42f8 100644 --- a/euclidean-spaces.tex +++ b/euclidean-spaces.tex @@ -1148,7 +1148,7 @@ $$ \begin{lemma}\label{lem:complex-unitary-1} Пусть $V$~--- унитарное пространство (внимание!), -$T\colon V\to V$~--- линейный оператора. +$T\colon V\to V$~--- линейный оператор. Предположим, что $B(T(v),v) = 0$ для всех $v\in V$. Тогда $T = 0$. \end{lemma} @@ -1198,7 +1198,7 @@ $$ \begin{remark} Замечание~\ref{rem:complex-unitary-counterexample} показывает, -что на эвклидовом пространстве оператор $T$ может удовлетворять +что на эвклидовом пространстве ненулевой оператор $T$ может удовлетворять тождеству $B(T(v),v)=0$ для всех $v\in V$. Однако, этого не может случиться для самосопряженного оператора. \end{remark} @@ -1257,7 +1257,7 @@ $||T(v)|| = ||T^*(v)||$ для всех $v\in V$. \begin{proof} Заметим, что оператор $T^*\circ T - T\circ T^*$ самосопряжен. По лемме~\ref{lem:selfadjoint-zero-characterisation} -равенство $T^*\circ T - T\circ T^*$ равносильно тому, +равенство $T^*\circ T - T\circ T^*$ нулю равносильно тому, что $B((T^*\circ T - T\circ T^*)(v),v) = 0$ для всех $v\in V$, что равносильно равенству $B(T^*(T(v)),v) = B(T(T^*(v)),v)$ для всех $v\in V$. @@ -1292,7 +1292,7 @@ $T$, соответствующие различным собственным ч По предложению~\ref{prop:normal-operator-adjoint-eigenvalues} теперь $T^*(u) = u\ol\lambda$. Поэтому $(\lambda-\mu)B(u,v) = B(u\ol\lambda,v) - B(u,v\mu) -= B(T(u),v) - B(u,T^*(v)) = 0$. += B(T^*(u),v) - B(u,T(v)) = 0$. Поскольку $\lambda\neq\mu$, из этого равенства следует, что $B(u,v)=0$, что и требовалось. \end{proof} @@ -1318,7 +1318,7 @@ $V$ диагональна. \begin{proof} Очевидно, что $(2)\Leftrightarrow(3)$ (см. также доказательство теоремы~\ref{thm:diagonalizable-equivalent}). -Покажем, что из (3) следует (1). Пусть матрица $t$ в некотором +Покажем, что из (3) следует (1). Пусть матрица $T$ в некотором ортонормированном базисе $\mc B$ диагональна. По предложению~\ref{prop:adjoint_matrix} матрица $T^*$ тогда получается из матрицы $T$ транспонированием @@ -1505,7 +1505,7 @@ $V$ диагональна. диагональна. Но диагональная матрица совпадает со своей транспонированной, поэтому $T=T^*$, откуда следует $(1)$. -Теперь мы докажем. что из $(1)$ следует $(2)$ индукцией по размености +Теперь мы докажем, что из $(1)$ следует $(2)$ индукцией по размерности пространства $V$. Если $\dim(V)=1$, утверждение очевидно. Пусть теперь $\dim(V) > 1$, и оператора $T$ самосопряжен. @@ -1518,7 +1518,7 @@ $V$ диагональна. и оператор $T|_{U^\perp}$ самосопряжен. По предположению индукции у $U^\perp$ есть ортонормальный базис, состоящий из собственных векторов оператора $T|_{U^\perp}$. -Присоединив к нему $u$, получаем ортонормальный базис $U^\perp$, +Присоединив к нему $u$, получаем ортонормальный базис $V$, состоящий из собственных векторов оператора $T$. \end{proof} @@ -1673,6 +1673,7 @@ $(2)\Rightarrow (1)$: несложно проверить, что матрица \literature{[F], гл. XIII, \S~5; [K2], гл. 3, \S~3, пп. 3, 6; [KM], ч. 2, \S~7, пп. 1--2, 4; \S~8, пп. 2--6.} +\nopagebreak Сейчас мы применим знания, полученные при изучении нормальных операторов, к некоторым частным случаям.