Конспект лекций по математическому анализу. Шерстнев А.Н. - 427 стр.

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   |lementy teorii neograni~ennyh
             operatorow

    x247. pONQTIE ZAMKNUTOGO OPERATORA
    1. lINEJNYM OPERATOROM (W DALXNEJ[EM PROSTO OPERATOROM) T W
GILXBERTOWOM PROSTRANSTWE H NAZYWAETSQ LINEJNOE OTOBRAVENIE
T : D(T ) ! H , GDE D(T ) | LINEAL W H (ON NAZYWAETSQ OBLASTX@ OPRE-
DELENIQ T ). oTMETIM, ^TO DLQ L@BOGO LINEJNOGO OPERATORA T = . oPE-
RATOR T NAZYWAETSQ PLOTNO ZADANNYM, ESLI LINEAL D(T ) PLOTEN W H .
lINEALY Ker T  ff 2 D(T )j Tf = g I R(T )  fTf j f 2 D(T )g NAZYWA-
@TSQ SOOTWETSTWENNO QDROM I OBRAZOM OPERATORA T .
    p R I M E R Y. 2. w GILXBERTOWOM PROSTRANSTWE H = L2[0; 1] OPREDELIM
OPERATOR M : (Mf )(t)  tf (t) (0  t  1); M OPREDELEN WS@DU W H I
OGRANI^EN.
    3. w GILXBERTOWOM PROSTRANSTWE H = L2(R) SNOWA POLOVIM (Mf )(t) 
tf (t) (t 2 R), GDE D(M ) = ff 2 L2(R)j tf (t) 2 L2(R)g; M PLOTNOZ ZADAN, NO
                                                                    n+1
NE OGRANI^EN: fn  [n;n+1] 2 D(M ), kfnk = 1, NO kMfn k2 = n t2 dt >
n2 (n 2 N). oPERATORY W PRIMERAH 2,3 NAZYWA@TSQ OPERATORAMI UMNOVE-
NIQ NA NEZAWISIMU@ PEREMENNU@.
    u P R A V N E N I Q. 4. pOKAVITE, ^TO T (f n)  (nf n ) | NEOGRANI^ENNYJ
PLOTNO ZADANNYJ LINEJNYJ OPERATOR W `2 .
    5. w GILXBERTOWOM PROSTRANSTWE L2 (R) POLOVIM (Tf )(t) = f 0 (t) (f 2
D(T )  D), GDE PROSTRANSTWO D OPREDELENO W 170.1. uBEDITESX, ^TO T |
NEOGRANI^ENNYJ PLOTNO ZADANNYJ LINEJNYJ OPERATOR.
    6. aLGEBRAI^ESKIE OPERACII NAD LINEJNYMI OPERATORAMI W GILXBER-
TOWOM PROSTRANSTWE H OPREDELQ@TSQ SOGLA[ENIQMI:
 D(A + B )  D(A) \ D(B );                       (A + B )f  Af + Bf ;
     D(A)  D(A);                                   (A)f  Af ( 2 C );
     D(AB )  ff 2 D(B )j Bf 2 D(A)g;               (AB )f  A(Bf ):


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