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In this sense, bounded operators are similar to finite-dimensional operators. The algebraic situation with unbounded operators is more involved because of generally different domains and ranges for different operators. zfO/ D zfO , 2 Dzf D Df . But the sum and product of two unbounded operators fO and gO with the respective domains Df and Dg are more involved. gO /; 2 Dfg D ˚ W « 2 Dg ; gO 2 Df : If fO or gO is defined everywhere, Df D H or Dg D H, then we respectively have Df Cg D Dg or Df Cg D Df .

Then fO D Á. For a continuous operator, the relation n ! 0 implies that fO n ! 0, and therefore, any continuous operator fO is closable, its closure fO is also continuous, and kfOk D kfOk. For future reference, the following assertion is formulated as a lemma. 23. , Df is either a closed subspace in H or the whole of H, and conversely, if Df D Df , a continuous operator fO is closed, fO D fO. It may be that … Df if fO is not closed; a continuous fO can be nonclosed, but is always closable (see below).

Then the direct sum is called the orthogonal direct sum. For such a sum, we use the sgn ˚, D D D1 ˚ D2 . This equality is equivalent to each of the equalities D1 D D « D2 and D2 D D « D1 . The notion of an orthogonal direct sum is easily extended to any number of mutually orthogonal subspaces: DD X ˚ Dk ; Dk ? Dl ; k ¤ l ; 8k; l: k 6 We are forced to useÂthe à symbol . 1 ž 2 / for a two-component column (“spinor”) instead of the conventional symbol 1 2 for reasons of space. 20 2 Linear Operators in Hilbert Spaces We call the operation ˚ of taking the orthogonal direct sum of (sub)spaces, ˚ D1 ; D2 !

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A game of aiming and evasion: General discussion and the marksman's strategies by Rufus Isaacs


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