Конспект лекций по математическому анализу. Шерстнев А.Н. - 431 стр.

U | UNITARNYJ OPERATOR (!!).
     eSLI T | PLOTNO ZADANNYJ OPERATOR W GILXBERTOWOM PROSTRANST-
WE H , TO ,(T ) = [U ,(T )]?.
  fg; gg 2 ,(T ) TTOGDA hTf; gi , hf; gi = 0 (f 2 D(T )) TTOGDA
hU ff; Tf g; fg; ggi = hfTf; ,f g; fg; ggi = hTf; gi,hf; gi = 0 (f 2 D(T ))
TTOGDA fg; gg 2 [U ,(T )]?: >
     4. uSTANOWIM SWOJSTWA SOPRQV      ENNOGO OPERATORA:
   (i) ESLI OPERATORY T; S PLOTNO ZADANY I S  T , TO T   S ,
  (ii) ESLI T | PLOTNO ZADANNYJ OPERATOR, TO T  ZAMKNUT.
 (iii) PLOTNO ZADANNYJ OPERATOR T ZAMYKAEM TTOGDA T  | PLOTNO ZA-
        DAN. pRI \TOM T = T .
 (iv) eSLI T PLOTNO ZADAN, TO H = Ker(T )  [R(T )],.
 (i). S  T ) (SM. 247.9) ,(S )  ,(T ) ) U ,(S )  U ,(T ) ) ,(S ) =
[U ,(S )]?  [U ,(T )]? = ,(T ) ) S   T .
     (ii). sLEDUET NEMEDLENNO IZ P. 3.
     (iii). oTMETIM, ^TO OPERATOR U IZ P. 3 UDOWLETWORQET RAWENSTWU
U 2 = ,I . sLEDOWATELXNO (SM. TAKVE 242.7),
         ,(T ), = ,(T )?? = [U 2,(T )]?? = [U (U ,(T ))?]? = [U ,(T )]?:
eSLI T  PLOTNO ZADAN, TO W SILU (ii) IZ DANNOGO RAWENSTWA SLEDUET, ^TO
LINEAL ,(T ) = ,(T ), | GRAFIK OPERATORA T , TAK ^TO T = T . oBRATNO,
PUSTX T  ZADAN NE PLOTNO. tOGDA NAJDETSQ g 2 D(T )? ; g 6= . tOGDA
fg; g 2 [,(T )]?, A ZNA^IT,
                   f; gg 2 U [,(T )? ] = [U ,(T )]? = ,(T ),:
|TO OZNA^AET, ^TO ,(T ), | NE GRAFIK, TO ESTX T NE ZAMYKAEM.
     (iv). g 2 R(T )? TTOGDA hTf; gi = 0 = hf; i (f 2 D(T )) TTOGDA (SM. P.
1) g 2 D(T ); T g =  TTOGDA g 2 Ker T : >
     u P R A V N E N I Q. 5. pOKAVITE, ^TO W USLOWIQH P. 1 g 2 D(T )
TTOGDA LINEJNYJ FUNKCIONAL f ! hTf; gi, ZADANNYJ NA LINEALE D(T ),
OGRANI^EN.
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