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Theory of Linear Operators in Hilbert Space (Dover Books on Mathematics), by N. I. Akhiezer
PDF Download Theory of Linear Operators in Hilbert Space (Dover Books on Mathematics), by N. I. Akhiezer
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Language Notes
Text: English (translation) Original Language: Russian
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Product details
Series: Dover Books on Mathematics
Paperback: 400 pages
Publisher: Dover Publications; First Thus edition (December 16, 1993)
Language: English
ISBN-10: 0486677486
ISBN-13: 978-0486677484
Product Dimensions:
5.4 x 0.8 x 8.4 inches
Shipping Weight: 15.2 ounces (View shipping rates and policies)
Average Customer Review:
3.8 out of 5 stars
5 customer reviews
Amazon Best Sellers Rank:
#1,163,845 in Books (See Top 100 in Books)
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This book has too my typo that makes it almost impossible to read. I hate to read a book like this.
The spectral theorem of David Hilbert, John von Neumann, and Marshall Stone gives a complete answer to the question of which operators admit a diogonal representation, up to unitary equivalence, and makes the question precise as well. The theorem states that these are the normal operators in Hilbert space. This includes the selfadjoint operators which represent observables in quantum physics, and the more interesting ones are unbounded. Remember the Heisenberg commutation relations do not admit bounded solutions. But there is a mathematical distinction between formally selfadjoint operators (also called symmetric operators) and the selfadjoint ones. It is only the latter to which the spectral theorem applies. The distinction between the two is understood from a pair of indicies (n,m), now called deficiency indices. In some applications they representboundary conditions, and when n = m, and the boundary conditions are assigned, the symmetric operator in question has selfadjoint extensions. And we know from von Neumann what they are. A central question in the book concerns the issue of unequal indices. Then selfadjoint extensions do not exist, at least not unless the Hilbert space is enlarged. A central theme in the book is that in case of unequal indices, there is a larger Hilbert space which does in fact admit selfadjoint extensions. The co-authors, along with Naimark, are the authorities on this. Because of applications to PDE theory and to physics, there has been constant interest in the theme right up to the present. Even the current interest, and lively activity, in quantum measurement theory (in connection with quantum information theory) and entanglement brings back to to the fore this old issue around diagonalizing operators by passing to an "enlarged" (or dilated)Hilbert space, or looking for an orthonormal basis in the extended Hilbert space. So the theme of the book is still current.
Increase background to better understand quantum computing fundamentals
great book for a back ground reading .....
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