Is there any branch of math that deals with functions or relations whose domains are operators?

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2
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I was thinking today.



I have this function, with this rule



$f:ℝ↦ℝ$



$x↦x+2$



this is



$f(x)=x+2 |x∈ℝ$



It follows elementarily that



$f(2)=2+2=4$



$f(3)=3+2=5$



and so on.



But what if I had a function with this rule



$s:+,-,×,÷↦ℝ$



$*↦frac12*frac35$



this is



$s(*)=frac12*frac35 | *∈+,-,×,÷ $



so



$s(+)=frac12+frac35=frac1110$



$s(-)=frac12-frac35=frac-110$



$s(×)=frac12×frac35=frac310$



$s(÷)=frac12÷frac35=frac56$



This would be a function since



enter image description here



And it's also biyective, so it has an inverse,



and I could do something like this:



$s^-1(frac56)=÷$



Now, Real analysis is the branch of math that deals with function whose domain is ℝ. Complex analysis is the branch of math that deals with function whose domain is ℂ.



So my question is, what is the branch of math that deals with functions whose domains are operators?



Thanks in advance.




algebra-precalculus functions soft-question




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  • 1




    Look at functional analysis, specifically when the domain of the operators is a space of functions from $mathbb R^2$ into $mathbb R,$ although the focus then would not be so much on the specific functions "addition" and "multiplication" and such but rather on all (continuous, differentiable, etc.) functions from $mathbb R^2$ into $mathbb R$. See also Is there a such thing as an operator of operators in mathematics?
    – Dave L. Renfro
    Jul 18 at 15:08















up vote
2
down vote

favorite












I was thinking today.



I have this function, with this rule



$f:ℝ↦ℝ$



$x↦x+2$



this is



$f(x)=x+2 |x∈ℝ$



It follows elementarily that



$f(2)=2+2=4$



$f(3)=3+2=5$



and so on.



But what if I had a function with this rule



$s:+,-,×,÷↦ℝ$



$*↦frac12*frac35$



this is



$s(*)=frac12*frac35 | *∈+,-,×,÷ $



so



$s(+)=frac12+frac35=frac1110$



$s(-)=frac12-frac35=frac-110$



$s(×)=frac12×frac35=frac310$



$s(÷)=frac12÷frac35=frac56$



This would be a function since



enter image description here



And it's also biyective, so it has an inverse,



and I could do something like this:



$s^-1(frac56)=÷$



Now, Real analysis is the branch of math that deals with function whose domain is ℝ. Complex analysis is the branch of math that deals with function whose domain is ℂ.



So my question is, what is the branch of math that deals with functions whose domains are operators?



Thanks in advance.




algebra-precalculus functions soft-question




share|cite|improve this question















  • 1




    Look at functional analysis, specifically when the domain of the operators is a space of functions from $mathbb R^2$ into $mathbb R,$ although the focus then would not be so much on the specific functions "addition" and "multiplication" and such but rather on all (continuous, differentiable, etc.) functions from $mathbb R^2$ into $mathbb R$. See also Is there a such thing as an operator of operators in mathematics?
    – Dave L. Renfro
    Jul 18 at 15:08













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I was thinking today.



I have this function, with this rule



$f:ℝ↦ℝ$



$x↦x+2$



this is



$f(x)=x+2 |x∈ℝ$



It follows elementarily that



$f(2)=2+2=4$



$f(3)=3+2=5$



and so on.



But what if I had a function with this rule



$s:+,-,×,÷↦ℝ$



$*↦frac12*frac35$



this is



$s(*)=frac12*frac35 | *∈+,-,×,÷ $



so



$s(+)=frac12+frac35=frac1110$



$s(-)=frac12-frac35=frac-110$



$s(×)=frac12×frac35=frac310$



$s(÷)=frac12÷frac35=frac56$



This would be a function since



enter image description here



And it's also biyective, so it has an inverse,



and I could do something like this:



$s^-1(frac56)=÷$



Now, Real analysis is the branch of math that deals with function whose domain is ℝ. Complex analysis is the branch of math that deals with function whose domain is ℂ.



So my question is, what is the branch of math that deals with functions whose domains are operators?



Thanks in advance.




algebra-precalculus functions soft-question




share|cite|improve this question











I was thinking today.



I have this function, with this rule



$f:ℝ↦ℝ$



$x↦x+2$



this is



$f(x)=x+2 |x∈ℝ$



It follows elementarily that



$f(2)=2+2=4$



$f(3)=3+2=5$



and so on.



But what if I had a function with this rule



$s:+,-,×,÷↦ℝ$



$*↦frac12*frac35$



this is



$s(*)=frac12*frac35 | *∈+,-,×,÷ $



so



$s(+)=frac12+frac35=frac1110$



$s(-)=frac12-frac35=frac-110$



$s(×)=frac12×frac35=frac310$



$s(÷)=frac12÷frac35=frac56$



This would be a function since



enter image description here



And it's also biyective, so it has an inverse,



and I could do something like this:



$s^-1(frac56)=÷$



Now, Real analysis is the branch of math that deals with function whose domain is ℝ. Complex analysis is the branch of math that deals with function whose domain is ℂ.



So my question is, what is the branch of math that deals with functions whose domains are operators?



Thanks in advance.





algebra-precalculus functions soft-question





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share|cite|improve this question




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asked Jul 18 at 7:28









Daniel Bonilla Jaramillo

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  • 1




    Look at functional analysis, specifically when the domain of the operators is a space of functions from $mathbb R^2$ into $mathbb R,$ although the focus then would not be so much on the specific functions "addition" and "multiplication" and such but rather on all (continuous, differentiable, etc.) functions from $mathbb R^2$ into $mathbb R$. See also Is there a such thing as an operator of operators in mathematics?
    – Dave L. Renfro
    Jul 18 at 15:08













  • 1




    Look at functional analysis, specifically when the domain of the operators is a space of functions from $mathbb R^2$ into $mathbb R,$ although the focus then would not be so much on the specific functions "addition" and "multiplication" and such but rather on all (continuous, differentiable, etc.) functions from $mathbb R^2$ into $mathbb R$. See also Is there a such thing as an operator of operators in mathematics?
    – Dave L. Renfro
    Jul 18 at 15:08








1




1




Look at functional analysis, specifically when the domain of the operators is a space of functions from $mathbb R^2$ into $mathbb R,$ although the focus then would not be so much on the specific functions "addition" and "multiplication" and such but rather on all (continuous, differentiable, etc.) functions from $mathbb R^2$ into $mathbb R$. See also Is there a such thing as an operator of operators in mathematics?
– Dave L. Renfro
Jul 18 at 15:08





Look at functional analysis, specifically when the domain of the operators is a space of functions from $mathbb R^2$ into $mathbb R,$ although the focus then would not be so much on the specific functions "addition" and "multiplication" and such but rather on all (continuous, differentiable, etc.) functions from $mathbb R^2$ into $mathbb R$. See also Is there a such thing as an operator of operators in mathematics?
– Dave L. Renfro
Jul 18 at 15:08
















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