Advanced functional analysis
41 videos • 1,761 views • by msc mathematics
1
Every λ ∈ C with │λ│› ││A││ is the regular point of the operator A
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2
ker(A-λI)=0 and im(A-λI)=x then λ is regular
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3
│λ│› ││A││ is regular point in C
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4
σ(A) is closed set in C
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5
Give an example of a continuous spectrum
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6
σ(A) is a closed set in c
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7
T is a compact operator then 0 ∈ σ(t)
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8
find an example for residual spectrum
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9
Define spectrum and classification of spectrum
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10
find the spectrum of right shift operator on L2
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11
λi is distinct eigen values, Ax=λx then {xi} are linearly independent
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12
Y ° ∈ E and ││y°││=1 then dist(y°, E¹)≥½
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13
Δλ=X then ker(Tλ)=0
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14
linearly independent eigen valuescorresponding to the eigen vectors are finite
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15
Δλ=Δλ closure
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16
minimax principle
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17
BANACH ALGEBRA
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18
examples of banach algebra
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19
p│E¹ =id E¹
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20
The operator Q=I-P is a projection and imP= ker Q and ker P = imQ
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21
let E1=Im p and E2= ker p, Then E1+E2 =E and E1∩ E2 =0(E1+E2 is a direct sum =E )
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22
P the projection onto E1 parallell to E2
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23
T:E→E linear. E1+E2 =E. P be projection onto E1||E2.Then PT=TP⟺ E1&E2 are invariant subspace of T
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24
if P is a projection and Im P orthogonal ker P, then P=P*
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25
Every orthoprojection P satisfies 0≤P≤I
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26
Gelfand's theorem on maximal ideals
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27
show that eigen values of a self adjoint operator are real
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28
Banach-steinhaus theorem
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29
define sub-linear functions on a linear space. Give an example which is not linear
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30
if A is a symmetric operator, show that ||A||=sup |‹Ax, x›|/||x²||
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31
define orthoprojection. Also show that a projection is an Orthoprojection if and only if ImP ⊥ Ker P
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32
state and prove Fredholm's first theorem
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33
Define schauder basis. Give an example
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34
if a positive I eigen value exist.then max(||x||=1)‹Tx, x›=λ1+
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35
strong cnvrgnc of sequence of operators.strong limit of a sequence of orthprjctn is an orthprjctn
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36
P be a projection on a hilbert space H. show that P is self adjoint if and only if ker( P )⊥ im( p )
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37
what is the difference between self adjoint operators and Ortho projection
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38
what is the difference between self adjoint operators and Ortho projection
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39
define closed graph operator and give an example
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40
Give an example of a set which is convex but is not perfectly convex
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41
find the spectrum (Kf)(t)=∫k(t, s)f(s)ds where k(t, s)=1,s≤t & 0,s›t
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