Algebre lineaire by Lipschitz S.

By Lipschitz S.

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By £1 we denote that subspace of 1{ whose basis elements are hj := 12j - 1 (j = 1,2, ... ) . - 12j - 1 + Tj12j (j = 1,2, ... ) . 1. DIRECT SUM OF SUBSPACES 53 Obviously, this sequence forms an orthogonal system. By £2 we denote the subspace with basis gi (j if f = 1,2, ... ) . The intersection of E £1 n £2 , then the vector f £1 and £2 consists of just the origin. Indeed, admits the representation 00 f= L1Jj(12i-1 +T 00 i l2i ) and f=L~iI2i-1' i=l i=1 The first equation yields (1, 12i) = 2- i T/j , while the second implies (1, 12i) = 0 (j = 1,2, ...

We shall prove that the operator Sf. - Sr fails to be compact in the space L2(r). Let t E f. 1 t E fl t E f 2. Sf. = -HrSrHr . Assuming that the operator Sf. - Sr is compact, we obtain that the operator T = X(Sr + HrSrHr ) is also compact, where X is the characteristic function of the curve f 2. 0 if t E f\[O, ~] and prove that we do not succeed in selecting a convergent subsequence from the sequence {Tcpn}r'. jii arc tan :2) ~ P + :2) = Cl n In (1 'tln (P;2) 2 arctan 1) ItG . and 1/Jn := ~Un+Vn.

3) Sr = -HrSrHr, where Hr is an operator defined in the space Lq(r,pl-q) by the relation (Hr

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