 About 225,000 results  books.google.com We shall see later that the density operator p describing states in statistical
quantum mechanics, as well as the product pA0 , where A0 is any bounded
observable, are completely continuous operators. The theory of completely
continuous ... 

 books.google.com The Concept of a Completely Continuous Operator Hilbert was the first to call
attention to an important class of operators that can be approximated by finite
dimensional operators, namely the completely continuous operators.
DEFINITION 3. 

 books.google.com If A is a completely continuous operator and if B is a bounded operator defined
everywhere in H, then the operators AB and BA are completely continuous. 2. If
A1 and A, are completely continuous operators, then 111 A1 + a._,A2 is a ... 

 books.google.com selfadjoint in if (Hu, v) = (u, Hv) for all u and v in the domain of H. We shall
discuss only selfadjoint operators. ... In other words, a completely continuous
operator transforms every bounded closed setin itsdomain into a compactset, to
which the ... 

 books.google.com In this chapter we define the notion of a completely continuous operator from a
Banach space to another Banach space and we present some simple properties
of the completely continuous operators. Next we prove Brouwer's fixed point ... 

 books.google.com But then the sequence (A1+A2)x;: is obviously also fundamental. I b. THEOREM.
The product (in either order) of a completely continuous operator A and a
bounded operator B is a completely continuous operator. Proof. Given any
bounded set ... 

 books.google.com An operator A is called completely continuous if it transforms bounded sets into
compact sets. Obviously a completely continuous operator is continuous, but the
converse is not necessarily true; even the identity operator on an ... 

 books.google.com THEOREM 4.2. a) Let A and B be two completely continuous linear operators on
X. Then any linear combination of A and B is a completely continuous operator,
too. b) If A is a completely continuous linear operator on X and C C L(X) then AC
... 

 books.google.com The points A of the continuous spectrum of the operator A that do not belong at
the same time to the point spectrum can be ... The identity operator E is
completely continuous if and only if the fundamental space H is finire
dimensionaL In this ... 

 books.google.com Prove that A is completely continuous if and only if {u,,} bounded implies {Au,,)
contains a strongly convergent subsequence. 2. Let A and B be completely
continuous operators on a Hilbert space H, and let T be a bounded operator.
Prove that ... 

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