Конспект лекций по математическому анализу. Шерстнев А.Н. - 432 стр.

   6.  pUSTX T | PLOTNO ZADAN, A S 2 B(H ). tOGDA (S + T ) = S  + T .
    7. pUSTX S; T; ST PLOTNO ZADANY. tOGDA (ST )  T S  . eSLI, W ^AST-
NOSTI, S 2 B(H ), TO (ST ) = T S .
    8. pUSTX T; T ,1 PLOTNO ZADANY. tOGDA (T ),1 = (T ,1 ).

    x249. |RMITOWY I SAMOSOPRQVENNYE OPERATORY
    1. pLOTNO ZADANNYJ OPERATOR T NAZYWAETSQ \RMITOWYM (ILI SIM-
METRI^ESKIM), ESLI T  T , ILI, ^TO \KWIWALENTNO,
                      hTf; gi = hf; Tgi (f; g 2 D(T )):
oPERATOR T NAZYWAETSQ SAMOSOPRQVENNYM, ESLI T = T .
    2. z A M E ^ A N I E. w SILU 248.4 (ii) WSQKIJ \RMITOW OPERATOR ZAMY-
KAEM.
    3. pUSTX T | SAMOSOPRQV      ENNYJ OPERATOR, A OPERATOR U | UNI-
TARNYJ. tOGDA OPERATOR UTU SAMOSOPRQVEN.
                                  
  iZ 248.7 SLEDUET, ^TO S  UTU  \RMITOW, T. E. S  S . oBRATNO (SNOWA
S U^ETOM 248.7),
      U SU = T = T  = (U (SU )) = (SU )U  U S U ) S  S : >
    p R I M E R Y. 4. [oPERATOR UMNOVENIQ NA NEZAWISIMU@ PEREMENNU@].
rASSMOTRIM W PROSTRANSTWE L2(R) OPERATOR M :
                  D(M )  ff 2 L2(R)j f () 2 L2(R)g;
                (Mf )()  f () ( 2 R; f 2 D(M )):
pOKAVEM, ^TO M | SAMOSOPRQVENNYJ OPERATOR. qSNO, ^TO M  M .
pUSTX g 2 D(M  ); g = M g. tOGDA DLQ WSEH f 2 D(M )
         Z                                     Z
           f ()g()d = hMf; gi = hf; g i = f ()g ()d )
                                          

                      Z
                        f ()[g() , g()]d = 0:


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